Control to output dynamic response and extend modulation index range with hybrid selective harmonic current mitigation-PWM and phase-shift PWM for four-quadrant cascaded H-bridge converters

ABSTRACT

A hybrid Cascaded H-Bridge (CHB) converter includes a selective harmonic current mitigation pulse width modulation (SHCM-PWM) unit coupled to an input current and providing an output signal SWSHCM, a phase shift pulse width modulation (PSPWM) unit coupled to the input current and providing an output signal SWPS, a modulation selector coupled to the output signal SWSHCM of the SHCM-PWM unit and the output signal SWPS of the PSPWM unit and providing an output signal SW, and a CHB converter coupled to the output signal SW of the modulation selector. The modulation selector can select one of the output signals (SWSHCM and SWPS) as the output signal SW based on the input current. The hybrid technique is for cascaded multilevel converters which utilizes asymmetric SHCM to mitigate the harmonics generated from PS-PWM to meet harmonic limits with a smaller number of switching transitions and smaller inductance than the conventional PS-PWM technique.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a Continuation-in-Part application of U.S.application Ser. No. 15/883,390 filed Jan. 30, 2018, which claimspriority to U.S. Application Ser. No. 62/454,997 filed Feb. 6, 2017, allof which are incorporated herein by reference in their entireties,including any figures, tables, and drawings.

GOVERNMENT FUNDING

This invention was made with government support under grant number1540118 awarded by the National Science Foundation. The government hascertain rights in the invention.

BACKGROUND

Multilevel power converters have drawn a lot of attention recently. Themodulation technique used in multilevel converters must have highefficiency, reduced passive filter cost, and fast transient responseunder different dynamic conditions. High efficiency is a critical metricfor multilevel converters. Because low switching frequencies lead to lowswitching power losses, low switching frequency modulation techniquessuch as selective harmonic elimination-PWM (SHE-PWM), selective harmonicmitigation-PWM (SHM-PWM), and selective harmonic current mitigation-PWM(SHCM-PWM) are promising to increase converter efficiencies. Inconventional SHE-PWM or SHM-PWM techniques, only the low order harmonicsare eliminated or mitigated to meet voltage harmonic limits. Hence, theconventional SHE-PWM and SHM-PWM techniques cannot ensure that currentharmonic limits are met, and these limits are more important than thevoltage harmonic limits for the grid tied converters. In addition, thegrid voltage harmonics can lead to unmitigated current harmonics forSHE-PWM and SHM-PWM techniques, but this information is not included inthe equations of these modulation techniques.

These two problems can be considered by introducing a SHCM-PWM techniquethat can meet the current harmonic limits of IEEE-519 by including theeffects of the grid voltage harmonics in the optimization process. Inthis technique, the coupling inductance between the converter and thegrid can be significantly reduced in comparison to SHE-PWM and SHM-PWMtechniques. Moreover, a higher number of current harmonics than SHE-PWMand SHM-PWM techniques can be mitigated with the same number ofswitching transitions. In He et al., based on the dynamic equations ofthe grid-tied converters, a high performance dynamic response can beachieved for a four-quadrant grid-tied converter. In addition, anindirect controller is used to change the active and reactive currentsfour times in each fundamental cycle. The modulation technique used inHe et al. is phase-shift PWM (PSPWM), which uses a high switchingfrequency to control low order harmonics. It is important to note thatthe SHCM-PWM technique could not be used with the indirect controllertechnique to obtain high dynamic performance. Because SHCM-PWM is anoffline modulation technique and the switching angles are calculated andstored in look-up tables, it needs to use fast Fourier transform (FFT),which results in time delays, to apply switching angles to theconverters. In addition, the number of switching transitions is very lowin SHCM-PWM, so it results in high ripple currents. As a result, it cancause intrinsic weak dynamic performance. When active or reactive powerare controlled with SHCM-PWM in four-quadrant converters, because theswitching angles need one fundamental cycle to get updated, a DC offsetremains on the injected currents for several cycles under dynamicconditions.

A new selective harmonic mitigation-pulse amplitude modulation (SHM-PAM)was proposed to eliminate the triplet harmonics of the CHB converter bycontrolling the switching angles and the DC-link voltages of cells ofthe CHB. Also, low-order non-triplet harmonics of the CHB voltage arecontrolled to meet the power quality voltage requirements. However, thistechnique needs to change all DC-link voltages of the CHB converter fordifferent modulation indices which can increase the complexity and thecost of the converter.

Recently, a fault-tolerant asymmetric selective harmonic elimination-PWM(asymmetric SHE-PWM) technique for the CHB inverter was proposed in togenerate a balanced AC voltage with the three-phase CHB converter whenone of the cells has a fault. A real-time selective harmonic eliminationtechnique is also proposed in to find the solutions of switching anglesof the low-frequency modulation technique in real-time. An indirectcontroller was proposed for having a transient-free dynamic responsewhen the active and reactive current of the grid-tied converter ischanged twice in a fundamental period. To reach this goal a highswitching frequency modulation technique (PS-PWM) was used to change theAC voltage of a grid-tied converter. So similar to using the PS-PWMtechnique in the transient period, the active and reactive current ischanged twice in a fundamental cycle. This leads to the lower speed ofchanging the AC current during dynamic conditions. So it is necessary tofind a single time instant to change the active and reactive current atthe same time. Also, the worst scenario for changing the active andreactive current is not discussed. Moreover, the effect of low-orderharmonics on the DC transient offset of the grid-tied converter for bothlow- and high-switching frequency were not discussed.

BRIEF SUMMARY

Embodiments of the subject invention provide novel and advantageoushybrid Cascaded H-Bridge (CHB) converters that selectively use aselective harmonic current mitigation pulse width modulation (SHCM-PWM)unit and a phase shift pulse width modulation (PSPWM) unit.

In an embodiment, a hybrid CHB converter can include a selectiveharmonic current mitigation pulse width modulation (SHCM-PWM) unitcoupled to an input current and providing an output signal SW_(SHCM), aphase shift pulse width modulation (PSPWM) unit coupled to the inputcurrent and providing an output signal SW_(PS), and a CHB converterselectively coupled to the SHCM-PWM and the PSPWM.

In another embodiment, a hybrid CHB converter can include a selectiveharmonic current mitigation pulse width modulation (SHCM-PWM) unitcoupled to an input current and providing an output signal SW_(SHCM), aphase shift pulse width modulation (PSPWM) unit coupled to the inputcurrent and providing an output signal SW_(PS), a modulation selectorcoupled to the output signal SW_(SHCM) of the SHCM-PWM unit and theoutput signal SW_(PS) of the PSPWM unit and providing an output signalSW, and a CHB converter coupled to the output signal SW of themodulation selector.

In yet another embodiment, a four-quadrant CHB converter can include aselective harmonic current mitigation pulse width modulation (SHCM-PWM)unit receiving an active power and a reactive power from a power grid, aphase shift pulse width modulation (PSPWM) unit receiving the activepower and the reactive power from the power grid, and a CHB converterselectively coupled to the SHCM-PWM unit at steady state and the PSPWMunit at transient state.

In another embodiment, the voltage harmonics due to the PS-PWM techniqueare mitigated with the harmonics generated from the low-frequencyasymmetric SHCM-PWM technique. Consequently, the switching frequency isreduced.

In another embodiment, the best and worst scenarios for changing theactive and reactive current of the grid-tied converter are derived.Using high-switching frequency modulation techniques such as PS-PWM canachieve a high-dynamic performance due to eliminating the low-orderharmonics and simplicity of controlling the fundamental and low-orderharmonics of the CHB.

In another embodiment, an asymmetric selective harmonic currentmitigation pulse width modulation (ASHCM-PWM) unit coupled to an inputcurrent and providing an output signal SW_(ASHCM); a phase shift pulsewidth modulation (PSPWM) unit coupled to the input current and providingan output signal SW_(PS); a P-cell H-Bridge coupled to the output signalSW_(ASHCM) of the ASHCM-PWM unit; and a N-cell H-Bridge coupled to theoutput signal SW_(PS) of the PSPWM unit.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 shows a configuration of a four-quadrant grid-tied CascadedH-Bridge (CHB) converter.

FIG. 2 shows a voltage waveform of an i-cell of a selective harmoniccurrent mitigation pulse width modulation (SHCM-PWM) converter.

FIG. 3 shows a required modulation index for a four-quadrant CHBconverter.

FIG. 4 shows a graph of switching angle solution vs. modulation indexwith SHCM-PWM technique.

FIG. 5 shows a phase diagram of a grid-tied converter.

FIG. 6 shows a hybrid CHB converter according to an embodiment of thesubject invention.

FIG. 7 shows a flowchart of a modulation selector block according to anembodiment of the subject invention.

FIG. 8 shows an indirect controller generating v_(ac-CHB).

FIG. 9 shows a harmonic spectrum of the v_(ac-CHB) with a phase shiftpulse width modulation (PSPWM) technique.

FIG. 10(a) shows a first simulation result of V_(ac-CHB), V_(ac-Grid),and I_(in) for a conventional four-quadrant converter.

FIG. 10(b) shows a first simulation result of V_(ac-CHB), V_(ac-Grid),and I_(in) for a four-quadrant converter according to an embodiment ofthe subject invention.

FIG. 10(c) shows a first simulation result of harmonic spectrum at 1000W−1000VAR for a four-quadrant converter according to an embodiment ofthe subject invention.

FIG. 10(d) shows a first simulation result of harmonic spectrum at 200W+250VAR for a four-quadrant converter according to an embodiment of thesubject invention.

FIG. 11(a) shows a second simulation result of V_(ac-CHB), V_(ac-Grid),and I_(in) for a conventional four-quadrant converter.

FIG. 11(b) shows a second simulation result of V_(ac-CHB), V_(ac-Grid),and I_(in) for a four-quadrant converter according to an embodiment ofthe subject invention.

FIG. 11(c) shows a second simulation result of harmonic spectrum at −850W+825VAR for a four-quadrant converter according to an embodiment of thesubject invention.

FIG. 11(d) shows a second simulation result of harmonic spectrum at −500W−600VAR for a four-quadrant converter according to an embodiment of thesubject invention.

FIG. 12 shows a hardware prototype of a four-quadrant CHB.

FIG. 13(a) shows a first experimental result of V_(ac-CHB), V_(ac-Grid),and I_(in) for a conventional four-quadrant converter.

FIG. 13(b) shows a first experimental result of V_(ac-CHB), V_(ac-Grid),and I_(in) for a four-quadrant converter according to an embodiment ofthe subject invention.

FIG. 13(c) shows a first experimental result of harmonic spectrum at1000 W−1000VAR for a four-quadrant converter according to an embodimentof the subject invention.

FIG. 13(d) shows a first experimental result of harmonic spectrum at 200W+250VAR for a four-quadrant converter according to an embodiment of thesubject invention.

FIG. 14(a) shows a second experimental result of V_(ac-CHB),V_(ac-Grid), and I_(in) for a conventional four-quadrant converter.

FIG. 14(b) shows a second experimental result of V_(ac-CHB),V_(ac-Grid), and I_(in) for a four-quadrant converter according to anembodiment of the subject invention.

FIG. 14(c) shows a second experimental result of harmonic spectrum at−850 W+825VAR for a four-quadrant converter according to an embodimentof the subject invention.

FIG. 14(d) shows a second experimental result of harmonic spectrum at−500 W−600VAR for a four-quadrant converter according to an embodimentof the subject invention.

FIG. 15(a) shows a grid-tied converter according to an embodiment of thesubject invention.

FIG. 15(b) shows a three-phase asynchronous motor according to anembodiment of the subject invention.

FIG. 15(c) shows a filter, according to an embodiment of the subjectinvention.

FIG. 16 shows a converter circuit, according to an embodiment of thesubject invention.

FIG. 17 shows a harmonic diagram, according to an embodiment of thesubject invention.

FIG. 18 shows a time domain waveform, according to an embodiment of thesubject invention.

FIG. 19 shows calculated inductance of harmonics, according to anembodiment of the subject invention.

FIG. 20 shows calculated inductance, according to an embodiment of thesubject invention.

FIGS. 21(a) and 21(b) show phase diagrams of h^(th) order harmonic,according to an embodiment of the subject invention.

FIG. 22 illustrates voltage harmonics, according to an embodiment of thesubject invention.

FIG. 23 shows calculated switching transitions in a half-period for aP-cell asymmetric SHCM-PW, according to an embodiment of the subjectinvention.

FIG. 24 shows solutions of the calculated phase and modulation index forthe disclosed PS-PWM and asymmetric SHCM-PWM technique in four-quadrantactive and reactive power operation, according to an embodiment of thesubject invention.

FIG. 25 shows solutions of the hybrid PS-PWM and asymmetric SHCM-PWMtechnique, according to an embodiment of the subject invention.

FIGS. 26(a)-(d) illustrate time-domain waveforms during dynamic responsefor the fundamental and 3^(rd) order of current harmonic, (a) bestscenario for 3^(rd) order current harmonic dynamic response, (b) worstscenario for 3^(rd) order current harmonic dynamic response, (c) bestscenarios of dynamic response for fundamental and 3^(rd) order currentharmonic, (d) worst scenarios of dynamic response for fundamental and3^(rd) order current harmonic, according to an embodiment of the subjectinvention.

FIG. 27(a) illustrates a control block diagram of, and FIG. 27(b) showsan indirect controller, of the hybrid PS-PWM and asymmetric SHCM-PWM,according to an embodiment of the subject invention.

FIGS. 28(a), (c) and (e) illustrate waveform in (a), harmonic spectra in(c), and the minimum inductance in (e) of a conventional PS-PWM.

FIGS. 28(b) and (d) illustrate waveform in (b) and harmonic spectra in(d) of PS-PWM and asymmetric SHCM-PWM technique, according to anembodiment of the subject invention.

FIGS. 29(a) and (c) illustrate waveform in (a) and harmonic spectra in(b), of a conventional PS-PWM.

FIGS. 29(b) and (d) illustrate waveform in (b) and harmonic spectra in(d) for a hybrid modulation technique, according to an embodiment of thesubject invention.

FIGS. 30(a), (b), (c) and (d) illustrate simulation results of thedynamic performance of the hybrid technique, (a) worst scenario, (b)best scenario, (c) current harmonic spectrum, and (d) current harmonicspectrum, according to an embodiment of the subject invention.

FIGS. 31(a) and (c) illustrate waveforms in (a) and harmonic spectra in(c), of a conventional PS-PWM.

FIGS. 31(b) and (d) illustrate waveforms in (b) and harmonic spectra in(d) for a hybrid modulation technique, according to another embodimentof the subject invention.

FIGS. 32(a) and (c) illustrate waveforms in (a) and harmonic spectra in(c), of a conventional PS-PWM.

FIGS. 32(b) and (d) illustrate waveforms in (b) and harmonic spectra in(d) for a hybrid modulation technique, according to yet anotherembodiment of the subject invention.

FIGS. 33(a), (b), (c) and (d) show experimental results of the dynamicperformance of the hybrid technique, (a) worst scenario, (b) bestscenario, (c) current harmonic spectrum, and (d) current harmonicspectrum, according to another embodiment of the subject invention.

DETAILED DESCRIPTION

Embodiments of the subject invention provide novel and advantageoushybrid Cascaded H-Bridge (CHB) converters that selectively use aselective harmonic current mitigation pulse width modulation (SHCM-PWM)unit under steady state and a phase shift pulse width modulation (PSPWM)unit under transient state.

The SHCM-PWM technique can be used in cascaded multilevel converters toextend the harmonic reduction spectrum, reduce the coupling inductanceand increase the efficiency. The offline SHCM-PWM technique has a lownumber of switching transitions, as its switching angles can only changeonce in a fundamental cycle, and relatively long time delays because ituses fast Fourier transform (FFT). As a result, its dynamic responseleaves much to be desired. For the four-quadrant power converters tohave good transient dynamic response, both active and reactive powermust be controlled at least two times in a fundamental cycle. Thus,embodiments of the subject invention can use a hybrid modulationtechnique including SHCM-PWM under steady state and PSPWM undertransient state. In addition, in order to extend the modulation indexrange and ensure that SHCM-PWM can process four-quadrant active andreactive power, the constraints of the switching angles for the SHCM-PWMcan be modified.

A hybrid modulation technique of an embodiment of the subject invention,which combines a SHCM-PWM technique and a PSPWM technique, is able toachieve high dynamic performance for four-quadrant grid-tied converters.Under steady state condition, the SHCM-PWM technique is applied toachieve high efficiency. Under dynamic condition, the PSPWM technique isemployed to update switching transitions several times in eachfundamental cycle to achieve high dynamic response performance.Furthermore, a controller is designed to switch between these twomodulations. In order to process four-quadrant active and reactivepower, the modulation index range of the SHCM-PWM can be greatlyextended by modifying the constraints of switching angles. The lowestnumber of switching transitions for the PSPWM technique is derived sothat it does not reduce efficiency and the performance of the indirectcontroller.

Embodiments of the subject invention can be applied to grid-tiedconverters. Also, embodiments of the subject invention can be used forrenewable energy sources, such as solar panels, to increase theefficiency and improve the dynamic performance. Moreover, embodiments ofthe subject invention can be used in ultra-fast charging stations ofelectrical vehicles to inject active and reactive powers to the grid.

In conventional selective harmonic elimination-PWM (SHE-PWM) or SHM-PWMtechniques, only the low order harmonics are eliminated or mitigated tomeet voltage harmonic limits. The conventional SHE-PWM and SHM-PWMtechniques cannot ensure that the current harmonic limits are met, andthese limits are more important than the voltage harmonic limits for thegrid tied converters. In addition, the grid voltage harmonics can leadto unmitigated current harmonics for SHE-PWM and SHM-PWM techniques, butthis information is not included in the equations of these modulationtechniques. These two problems can be solved by introducing a SHCM-PWMtechnique. Hybrid modulation techniques of embodiments of the subjectinvention are able to achieve high dynamic performance for four-quadrantgrid-tied converters by combining a SHCM-PWM technique and a PSPWMtechnique. Under steady state condition, the SHCM-PWM technique isapplied in order to achieve high efficiency, and under dynamiccondition, the PSPWM technique is applied to update switchingtransitions several times in each fundamental cycle to achieve highdynamic response performance. Further, a controller can be provided toswitch between these two modulations. In order to process four-quadrantactive and reactive power, the modulation index range of SHCM-PWM can begreatly extended by modifying the constraints of switching angles. Thelowest number of switching transitions for PSPWM technique is derived sothat it does not reduce efficiency and the performance of the indirectcontroller.

Grid-tied four-quadrant converters need large modulation index range towork with different active and reactive loads [8]. The modulation indexrange of low frequency modulation techniques depends on optimizationconstraints applied to the Fourier series equations. To increasemodulation index range of low frequency modulation techniques, either anunequal DC link voltage technique or modified switching angleconstraints can be used.

However, a hybrid modulation technique of an embodiment of the subjectinvention, which combines SHCM-PWM and PSPWM, achieves high dynamicperformance for four-quadrant grid-tied converters, because the SHCM-PWMtechnique is applied under steady state to achieve high efficiency andthe PSPWM technique is applied under dynamic condition to updateswitching transitions several times in each fundamental cycle to achievehigh dynamic response performance. Further, a controller can be designedto selectively switch between these two modulations.

FIG. 1 shows a configuration of a four-quadrant CHB grid-tied converter.Referring to FIG. 1, the CHB converter is connected to a power grid withcoupling inductance L_(F) and parasitic resistance R_(F). The CHBconverter includes i number of cells. The DC link voltages are equal toV_(dc), and the DC side of each cell is directly connected to theisolated DC/DC converters. The outputs of isolated DC/DC converters arein parallel to charge energy storage on a DC bus. The loads can beconnected to the DC bus with bi-directional DC/DC converters in FIG. 1.Because DC link voltages of the CHB converter can be regulated with theisolated DC/DC converters, the DC links of embodiments of the subjectinvention can be connected to DC sources. The time domain currentequation of the CHB converter on AC side is,

$\begin{matrix}{{{v_{{{ac} - {CHB} - {h{(t)}}} = L_{F}}\frac{{di}_{{in} - h}(t)}{dt}} + {R_{F}{i_{{in} - h}(t)}} + {v_{{ac} - {Grid} - h}(t)}},} & (1)\end{matrix}$

In equation (1), v_(ac-Grid-h), v_(ac-CHB-h), and i_(in-h) are theh^(th) harmonic order of the grid voltage, CHB voltage, and injectedcurrent, respectively. The relationship of fundamental frequency (60/50Hz) component v_(ac-Grid-1), v_(ac-CHB-1), and i_(in-1) can be obtainedin equation (1). The quarter period waveform of v_(ac-CHB-h) for thei-cell CHB converter in FIG. 1, when j^(th) cell has n_(j) (j=1, 2 . . ., i) switching angles in each quarter period, is shown in FIG. 2. FIG. 2shows a voltage waveform of an i-cell of a SHCM-PWM converter. Due toquarter wave symmetry, the Fourier series equations of FIG. 2 can bewritten as,

$\begin{matrix}{\left. {{{v_{{ac} - {CHB}}(t)} = {\sum\limits_{h = 1}^{\infty}{\frac{4V_{dc}}{\pi\; h}b_{h}\mspace{20mu}{\sin\left( {h\;\omega\; t} \right)}}}},{b_{h} = {\left( {{\cos\left( {h\;\theta_{11}} \right)} - {\cos\mspace{14mu} h\;\theta_{12}}} \right) + \;{.\;.\;.\mspace{14mu}{+ \;{\cos\left( {h\;\theta_{i{(n_{i})}}} \right)}}}}}} \right),} & (2)\end{matrix}$where θ₁₁, θ₁₂, . . . , θ_(i(n) _(i) ₎ are the switching angles of theCHB converter in each quarter period as shown in FIG. 2.4V_(dc)b_(h)/(πh) is the magnitude of the h^(th) order harmonic forv_(ac-CHB)(t). When h=1, the modulation index (M_(a)=b₁) of the CHBconverter is obtained from equation (2).

The power quality standard that is used to meet both current and voltageharmonics is IEEE 519. The limits of both current and voltage harmonicsat the point of common coupling (PCC) are provided in Table I below. InIEEE-519, I_(L) is the maximum demand load current of the four-quadrantconverter. I_(sc) is the short circuit current at the PCC.

TABLE I CURRENT AND VOLTAGE HARMONIC LIMITS OF IEEE 519 STANDARD(I_(sc)/I_(L) ≤ 20) [11] FOR GRID VOLTAGE LESS THAN 69 κV. Currentharmonics and Voltage harmonics and Harmonic total demand distortiontotal harmonic order (h) TDD distortion THD  3 ≤ h < 11   4% 3% 11 ≤ h <17   2% 3% 17 ≤ h < 23 1.5% 3% 23 ≤ h < 35 0.6% 3% 35 ≤ h 0.3% 3% TDD orTHD   5% 5%

The key parameters, such as the switching frequency of each switch, thenumber of harmonics that can be mitigated with the SHCM-PWM, and thecoupling inductance between the converter and the grid, can beconsidered. When the grid voltage harmonics (|V_(ac-Grid-h)|) have thehighest magnitudes under the worst scenario defined in Table I, theequation set that is used to find the solutions of SHCM-PWM to meetcurrent harmonic limits of IEEE 519 in Table I is shown below,

$\begin{matrix}\left\{ \begin{matrix}{{M_{a} = {{\cos\mspace{20mu}\theta_{11}} - {\cos\mspace{20mu}\theta_{12}} + {\cos\mspace{20mu}\theta_{13}} + \;{{.\;.\;.\mspace{14mu}{+ \;\cos}}\mspace{14mu}\theta_{K}}}},} \\{{\frac{{v_{{ac} - {Grid} - h}{{+ {v_{{ac} - {CHB} - h}}}}}}{{\omega\;{hL}_{T}I_{L}}} \leq C_{h}},{h = 3},5,{7\mspace{14mu}.\;.\;.}} \\{{\sqrt{\left( \frac{I_{{in} - 3}}{I_{L}} \right)^{2} + \left( \frac{I_{{in} - 5}}{I_{L}} \right)^{2} + \;{.\;.\;.\mspace{14mu}{+ \left( \frac{I_{{in} - h}}{I_{L}} \right)^{2}}}} \leq C_{TDD}},}\end{matrix} \right. & (3)\end{matrix}$where K is the number of switching transitions of the SHCM-PWM during aquarter fundamental period (K=n₁₁+n₁₂+ . . . n_(i(ni))), and C_(h) andC_(TDD) are the current harmonics and TDD limits of i_(in) in Table I.By using guidelines in Moeini et al. [6], which is hereby incorporatedby reference herein in its entirety, the parameters can be calculated asshown in Table II.

TABLE II CALCULATED CIRCUIT PARAMETERS OF SHCM-PWM TECHNIQUE ParameterSymbol Value Line frequency F 60 Hz AC grid Voltage (RMS) V_(ac-Grid)110 V Total rated power S_(total) 1.5 kVA Maximum Demand Load I_(L)14.14 A (RMS) Number of H-bridge cells i  3 Number of switching K  9transitions Highest order of mitigated H 69^(th) harmonic in (3) DC busvoltage V_(dc) 73 V Coupling inductance L_(F) 10 mH (0.485 p.u.)Parasitic resistance of L_(F) R_(F) 0.6 Ω

To ensure that the SHCM-PWM modulation technique can properly work insteady state for four-quadrant active and reactive power, thelimitations for the maximum and minimum modulation indices can beobtained based on equation (1). In equation (1), the modulation index ofCHB voltage is,

$\begin{matrix}{M_{a} = {{\frac{\pi}{4V_{dc}}\left( {{V_{{ac} - {Grid} - 1}{\angle 0}} + {\left( {{j\;\omega\; L_{F}} + R_{F}} \right)I_{{in} - 1}{\angle\theta}_{I_{{in} - 1}}}} \right)}}} & (4)\end{matrix}$

FIG. 3 shows a required modulation index for a four-quadrant CHBconverter. In equation (4), if the grid voltage is taken as thereference, by changing the magnitude and phase of i_(in-1)(0<I_(in-1)<I_(L), 0≤θ_(I) _(in-1) <2π), the required modulation indicesare derived as FIG. 3, for the circuit parameters in Table II. Referringto FIG. 3, the CHB converter can process four-quadrant active andreactive power at steady state when modulation index changes from 0.85to 2.485.

The conventional constraints of the switching angles used to solveswitching angles for the equation set in equation (3) are,

$\begin{matrix}{0 < \theta_{11} < \theta_{12} < \;{.\;.\;.}\mspace{14mu} < \theta_{{in}_{i}} < \frac{\pi}{2}} & (5)\end{matrix}$

The constraints in equation (5) undesirably restrict the optimizationtechniques used to solve equation (3). The switching angle solutionrange of SHCM-PWM technique can be significantly improved by modifyingthe constraints to,

$\begin{matrix}{{0 < \theta_{11} < \frac{\pi}{2}},{0 < \theta_{12} < \frac{\pi}{2}},\;{.\;.\;.}\mspace{14mu},{0 < \theta_{{in}_{i}} < \frac{\pi}{2}}} & (6)\end{matrix}$

The multi-objective particle swarm optimization (MOPSO) technique, as inReyes-Sierra et al. [12] (which is hereby incorporated by referenceherein in its entirety), can be used to solve equation (3). Themodulation index ranges of using the switching angle constraints inequations (5) and (6) for equation (3) are compared in FIG. 4. FIG. 4shows a graph of switching angle solution vs. modulation index with aSHCM-PWM technique. Referring to FIG. 4, the switching angle solutionwith conventional constraints in equation (5) limits the modulationindex to [1.78, 2.495]. The modulation index is greatly extended to[0.8, 2.495] with the modified switching angle constraints in equation(6); it covers all of the required modulation indices in FIG. 3.

By changing the modulation index, the magnitude of v_(ac-CHB-1) inequation (2) can be controlled. However, in order to track desiredactive and reactive power for four-quadrant operations, the phase of CHBvoltage should also be controlled. Because of this, if the phase of theCHB voltage is θ and 0<θ<2π, (2) can be rewritten as,

$\begin{matrix}{{v_{{ac} - {CHB}}(t)} = {\sum\limits_{h = 1}^{\infty}{\frac{4V_{dc}}{\pi\; h}b_{h}\mspace{14mu}{\sin\left( {{h\;\omega\; t} + {h\;\theta}} \right)}}}} & (7) \\{{or},} & \; \\{{v_{{ac} - {CHB}}(t)} = {\sum\limits_{h = 1}^{\infty}{\frac{4V_{dc}}{\pi\; h}{b_{h}\left( {{{\cos\left( {h\;\theta} \right)}{\sin\left( {h\;\omega\; t} \right)}} + {{\sin\left( {h\;\theta} \right)}{\cos\left( {h\;\omega\; t} \right)}}} \right)}}}} & (8)\end{matrix}$

Because when the phases of both I_(in-1) and V_(ac-CHB) change from 0 to2π, there are switching angle solutions, the CHB can handlefour-quadrant active and reactive power.

FIG. 5 shows a phase diagram of a grid-tied converter. The dq phasordiagram of equation (1) for the fundamental frequency is shown in FIG.5.

The dq frame rotates counterclockwise with speed □. The injected currenti_(in)(t) is composed of dq components as,i _(in)(t)=I _(in-d) sin(ωt)+I _(in-q) cos(ωt)  (9)From FIG. 5, the CHB voltage is,v _(ac-CHB)(t)=V _(ac-CHB-d) sin(ωt)+V _(ac-CHB-q) cos(ωt),V _(ac-CHB-d) =−L _(F) ωI _(in-q) +R _(F) I _(in-d) +V _(ac-Grid),V _(ac-CHB-q) =L _(F) ωI _(in-d) +R _(F) I _(in-q),  (10)

In order to have the desired current in FIG. 5, the CHB voltage can becontrolled with b₁ and θ in equation (8) by using the followingequations,

$\begin{matrix}{{V_{{ac} - {CHB} - d} = {\frac{4V_{dc}b_{1}}{\pi}{\cos(\theta)}}},{V_{{ac} - {CHB} - q} = {\frac{4V_{dc}b_{1}}{\pi}{{\sin(\theta)}.}}}} & (11)\end{matrix}$

In the time domain, if the changes of dq current references causev_(ac-CHB)(t) to change by Δ_(vac-CHB)(t) from v_(ac_CHB1)(t) tov_(ac_CHB2)(t), and i_(in)(t) to change by Δi_(in)(t) from i_(in1)(t) toi_(in2)(t), the following equations hold,i _(in2)(t)=i _(in1)(t)+Δi _(in)(t),v _(ac-CHB2)(t)=v _(ac-CHB1)(t)+Δv _(ac-CHB)(t),  (12)

It is assumed that the grid voltage does not change under the transientcondition so Δ_(vac-Grid)=0. Based on FIG. 5, Δi_(in) and Δ_(vac-CHB)can be derived as,ΔV _(ac-CHB-d) =−L _(F) ωΔI _(in-q) +R _(F) ΔI _(in-d),ΔV _(ac-CHB-q) =L _(F) ωΔI _(in-d) +R _(F) ΔI _(in-q),Δv _(ac-CHB)(t)=ΔV _(ac-CHB-d) sin(ωt)+ΔV _(ac-CHB-q) cos(ωt)Δi _(in)(t)=ΔI _(in-d) sin(ωt)+ΔI _(in-q) cos(ωt)  (13)

The differential equation under transient duration is,

$\begin{matrix}{{\Delta\;{v_{{ac} - {CHB}}(t)}} = {{L_{F}\frac{d\;\Delta\;{i_{in}(t)}}{dt}} + {R_{F}\Delta\;{i_{in}(t)}}}} & (14)\end{matrix}$

If the current changes at t=t0, from equations (13) and (14), theΔi_(in) can be solved as,

$\begin{matrix}{{\Delta\;{i_{in}(t)}} = {{ce}^{\frac{- R_{F}}{L_{F}}t} + \left( {{\Delta\; i_{{in} - d}\mspace{11mu}{\sin\left( {\omega\; t} \right)}} + {\Delta\; I_{{in} - q}\mspace{11mu}{\cos\left( {\omega\; t} \right)}}} \right)}} & (15)\end{matrix}$where c depends on both Δv_(ac-CHB) and the initial condition ofΔi_(in). If the control signal of ΔI_(in-d) and ΔI_(in-q) change at t=t₀and Δi_(in)(t0-)=0, c can be derived as,

$\begin{matrix}{c = {- {e^{\frac{R_{F}}{L_{F}}t_{0}}\left( {{\Delta\; i_{{in} - d}\mspace{11mu}{\sin\left( {\omega\; t_{0}} \right)}} + {\Delta\; I_{{in} - q}\mspace{11mu}{\cos\left( {\omega\; t_{0}} \right)}}} \right)}}} & (16)\end{matrix}$

The second term in equation (15) is the steady state term of Δi_(in).The first term in equation (15) is an undesirable transient current. Inorder to remove undesirable transient current, in equation (16), cshould be always equal to zero. Because of this, ΔI_(in-d) or ΔI_(in-q)should only change when sin(ωt₀) or cos(ωt₀) are equal to zero. Thisindicates if dq currents in equation (16) change under the followingconditions, i_(in) will have no transient currents.

$\begin{matrix}\left\{ \begin{matrix}{{{\omega\; t_{0}} = {k\;\pi}},} & {\Delta\; I_{{in} - d}\mspace{14mu}{should}\mspace{14mu}{change}} \\{{{\omega\; t_{0}} = {{k\;\pi} + \frac{\pi}{2}}},} & {\Delta\; I_{{in} - q}\mspace{14mu}{should}\mspace{14mu}{change}}\end{matrix} \right. & (17)\end{matrix}$

To have the fast transient response in practice, the active power, whichis determined by ΔI_(in-d), and reactive power, which is determined byΔI_(in-q), must change at times defined in equation (17). Therefore, thecurrents can have 2 to 4 changes within one cycle. At the same time, theexisting technique uses only the PSPWM technique to improve thetransient condition. However, the mitigation of low order currentharmonics using the PSPWM technique needs more switching transitionsthan low frequency modulation techniques such as SHE-PWM. As a result,the PSPWM technique has a high switching power loss. Hybrid SHCM-PWM andPSPWM techniques of embodiments of the subject invention solve theseissues. In embodiments of the subject invention, the SHCM-PWM techniqueis employed under the steady state condition and the PSPWM technique isemployed under the transient condition.

FIG. 6 shows a hybrid CHB converter according to an embodiment of thesubject invention. The block diagram in FIG. 6 shows the hybrid SHCM-PWMand PSPWM technique. In this embodiment, when current referencesΔI*_(in-d) and ΔI*_(in-q) change, the following conditions must be usedby the modulation selector to select the modulation technique for theCHB converter,

If |ΔI*_(in-d)|>0 & ωt=kπ, use PSPWM (SW_(PS)) until ωt=(k+2)π If|ΔI*_(in-q)|>0 & ωt=kπ+π/2, use PSPWM (SW_(PS)) until ωt=(k+2)πOtherwise, use SHCM-PWM.

FIG. 7 shows a flowchart of a modulation selector block according to anembodiment of the subject invention, and FIG. 8 shows an indirectcontroller generating v_(ac-CHB). The flowchart for the modulationselector block in FIG. 6 is shown in FIG. 7. The block diagram of theindirect controller based on equations (12), (13), and (17) is shown inFIG. 8. The output of the indirect controller in FIG. 8 is v_(ac-CHB2).Because the PSPWM technique does not use FFT to change v_(ac-CHB), it ispossible to change v_(ac-CHB) several times in a fundamental period. Onthe other hand, the SHCM-PWM technique needs to use the FFT block toobtain the modulation index M_(a), which is needed for checking look uptables and changing the output voltage of the CHB converter. Because theFFT block has time delays, the SHCM-PWM technique needs at least onecycle to change v_(ac-CHB). However, equation (17) requires changing thev_(ac-CHB) at least twice in a fundamental period so both active andreactive power can be controlled for a four-quadrant grid-tiedconverter. Therefore, the PSPWM technique is an appropriate techniquefor dynamic response improvement.

The switching frequency of the PSPWM technique must be designed to haveboth good dynamic response and low switching power loss. To reduce theswitching power loss, the switching frequency of PSPWM must be chosen aslow as possible when the dynamic response is greatly improved. However,reducing the switching frequency of PSPWM may lead to undesirably-highlow-order voltage harmonics, which includes the fundamentalv_(ac-CHB-1). Because of this, the lowest PSPWM switching frequency,which does not affect v_(ac-CHB-1), can be explored. The output voltageof an i-cell CHB converter with the PSPWM technique can be written as,

$\begin{matrix}{{v_{{ac} - {CHB} - {PSPWM}}(t)} - {i\; V_{dc}M\mspace{14mu}{\cos\left( {{\omega_{0}t} + \theta_{0}} \right)}} + {\frac{4V_{dc}}{\pi}{\sum\limits_{B = 1}^{\infty}{\sum\limits_{A = {- \infty}}^{\infty}\left( {\frac{1}{2B}{J_{{2A} - 1}\left( {{iB}\;\pi\; M} \right)} \times {\sin\left( {\left( {{2{iB}} + {2A} - 1} \right)\frac{\pi}{2}} \right)}{\cos\left( {{2{iB}\;\omega_{c}t} + {\left( {{2A} - 1} \right)\left( {{\omega_{0}t} + \theta_{0}} \right)}} \right)}} \right)}}}} & (18)\end{matrix}$

where, ω₀=2πf₀, f₀ and θ₀ are the fundamental frequency and phase of CHBvoltage. M is the modulation index of each cell of CHB. The totalmodulation index M_(a) of the CHB converter is iM. ω_(c)=2πf_(c) andf_(c) is the average carrier frequency of each cell. B is the baseband,and A is the sideband harmonics of each baseband harmonic as shown inFIG. 9. FIG. 9 shows the harmonic spectrum of the v_(ac-CHB) with PSPWMtechnique. J is the Bessel function of first kind. The bandwidth ofB^(th) baseband harmonic in FIG. 9 can be obtained with the followingequation,BW _(B)≈2(iMBπ+2)f ₀  (19)

In equation (18), the switching frequency fs of the v_(ac-CHB) withPSPWM is equal to 2if_(c). In order not to generate sideband harmonicsoverlapping and influencing v_(ac-CHB-1), the carrier frequency of theCHB converter for the first baseband (B=1) can be derived based onfollowing equation,f _(z) −BW ₁ >f ₀ ⇒f _(s) >f ₀ +BW ₁ ⇒f _(s)>(2iMBπ+5)f ₀  (20)

Based on equation (20), when i=3, M=1 (the maximum modulation index foreach cell) and in the worst scenario, the lowest f_(s) and f_(c) aretherefore 1440 Hz and 240 Hz respectively.

The subject invention includes, but is not limited to, the followingexemplified embodiments.

Embodiment 1

A hybrid Cascaded H-Bridge (CHB) converter, comprising:

a selective harmonic current mitigation pulse width modulation(SHCM-PWM) unit coupled to an input current and providing an outputsignal SW_(SHCM);

a phase shift pulse width modulation (PSPWM) unit coupled to the inputcurrent and providing an output signal SW_(PS); and

a CHB converter selectively coupled to the SHCM-PWM unit and the PSPWMunit.

Embodiment 2

The hybrid CHB converter according to embodiment 1, wherein the CHBconverter is coupled to the SHCM-PWM unit under steady state conditionand the CHB converter is coupled to the PSPWM unit under dynamiccondition.

Embodiment 3

The hybrid CHB converter according to embodiment 2, wherein the CHBconverter is coupled to the PSPWM unit under transient condition.

Embodiment 4

The hybrid CHB converter according to any of embodiments 2-3, whereinthe input current includes an active current reference ΔI*_(in-d) and areactive current reference ΔI*_(in-q), and the CHB converter isselectively coupled to the SHCM-PWM unit and the PSPWM unit based on theactive current reference ΔI*_(in-d) and the reactive current referenceΔI*_(in-q).

Embodiment 5

The hybrid CHB converter according to embodiment 4, wherein the CHBconverter is selectively coupled to the PSPWM unit in case the inputcurrent satisfies the following Formula 1:|ΔI* _(in-d)|>0 & ωt=kπ, until ωt=(k+2)π.  Formula 1

Embodiment 6

The hybrid CHB converter according to embodiment 5, wherein the CHBconverter is selectively coupled to the PSPWM unit in case the inputcurrent satisfies the following Formula 2:|ΔI* _(in-q)|>0 & ωt=kπ+π/2, until ωt=(k+2)π.  Formula 2

Embodiment 7

The hybrid CHB converter according to embodiment 6, wherein the CHBconverter is selectively coupled to the SHCM-PWM unit in all cases wherethe input current does not satisfy either of Formula 1 and Formula 2.

Embodiment 8

The hybrid CHB converter according to any of embodiments 2-7, furthercomprising an indirect controller coupled to the input current andproviding an output current v_(ac-CHB2) to the SHCM-PWM unit and thePSPWM unit.

Embodiment 9

A hybrid Cascaded H-Bridge (CHB) converter, comprising:

a selective harmonic current mitigation pulse width modulation(SHCM-PWM) unit coupled to an input current and providing an outputsignal SW_(SHCM);

a phase shift pulse width modulation (PSPWM) unit coupled to the inputcurrent and providing an output signal SW_(PS);

a modulation selector coupled to the output signal SW_(SHCM) of theSHCM-PWM unit and the output signal SW_(PS) of the PSPWM unit andproviding an output signal SW; and

a CHB converter coupled to the output signal SW of the modulationselector.

Embodiment 10

The hybrid CHB converter according to embodiment 9, wherein themodulation selector is connected to the input current.

Embodiment 11

The hybrid CHB converter according to embodiment 10, wherein themodulation selector selects one of the output signal SW_(SHCM) and theoutput signal SW_(PS) as the output signal SW based on the inputcurrent.

Embodiment 12

The hybrid CHB converter according to embodiment 11, further comprisingan indirect controller coupled to the input current and providing anoutput current v_(ac-CHB2) to the SHCM-PWM unit and the PSPWM unit.

Embodiment 13

The hybrid CHB converter according to embodiment 12, wherein the inputcurrent includes an active current reference ΔI*_(in-d) and a reactivecurrent reference ΔI*_(in-q), and the modulation selector selects one ofthe output signal SW_(SHCM) and the output signal SW_(PS) based on theactive current reference ΔI*_(in-d) and the reactive current referenceΔI*_(in-q).

Embodiment 14

The hybrid CHB converter according to embodiment 13, wherein themodulation selector selects the output signal SW_(PS) in case the inputcurrent satisfies the following Formulas 3 and 4:|ΔI* _(in-d)|>0 & ωt=kπ, until ωt=(k+2)π,  Formula 3|ΔI* _(in-q)|>0 & ωt=kπ+π/2, until ωt=(k+2)π.  Formula 4

Embodiment 15

The hybrid CHB converter according to embodiment 14, the modulationselector selects the output signal SW_(SHCM) in all cases where theinput current does not satisfy both Formula 3 and Formula 4.

Embodiment 16

The hybrid CHB converter according to any of embodiments 12-15, furthercomprising a phase lock loop (PLL) coupled to the modulation selectorand an output of the CHB converter.

Embodiment 17

The hybrid CHB converter according to any of embodiments 9-16, whereinswitch angles of the CHB converter are modified such that each of theswitch angles has a range of 0 to π/2 (alternatively, or in addition,each of the switch angles is in a range of 0 to π/2).

Embodiment 18

The hybrid CHB converter according to embodiment 17, wherein the switchangles of the CHB are calculated and stored in a look up table.

Embodiment 19

The hybrid CHB converter according to embodiment 18, wherein theSHCM-PWM unit uses a FFT block.

Embodiment 20

The hybrid CHB converter according to embodiment 19, wherein theSHCM-PWM unit obtains a modulation index for checking the look up table.

Embodiment 21

A four-quadrant Cascaded H-Bridge (CHB) converter, comprising:

a selective harmonic current mitigation pulse width modulation(SHCM-PWM) unit receiving an active power and a reactive power from apower grid;

a phase shift pulse width modulation (PSPWM) unit receiving the activepower and the reactive power from the power grid; and

a CHB converter selectively coupled to the SHCM-PWM unit at steady stateand the PSPWM unit at transient state.

Embodiment 22

The four-quadrant CHB converter according to embodiment 21, wherein theactive power and the reactive power are changed separately within onecycle.

Embodiment 23

The four-quadrant CHB converter according to any of embodiments 21-22,wherein a switching frequency of the PSPWM unit is 240 Hertz (Hz).

Embodiment 24

The four-quadrant CHB converter according to any of embodiments 21-23,wherein a modulation index with the SHCM-PWM unit is in a range of 0.8to 2.495.

Embodiment 25

A grid-tied converter, comprising:

a selective harmonic current mitigation pulse width modulation(SHCM-PWM) unit coupled to an input current and providing an outputsignal SW_(SHCM);

a phase shift pulse width modulation (PSPWM) unit coupled to the inputcurrent and providing an output signal SW_(PS);

a modulation selector coupled to the output signal SW_(SHCM) of theSHCM-PWM unit and the output signal SW_(PS) of the PSPWM unit andproviding an output signal SW; and

a H bridge converter coupled to the output signal SW of the modulationselector.

Embodiment 26

The grid-tied converter according to embodiment 25, wherein themodulation selector is connected to the input current.

Embodiment 27

The grid-tied converter according to any of embodiments 25-26, whereinthe modulation selector selects one of the output signal SW_(SHCM) andthe output signal SW_(PS) as the output signal SW based on the inputcurrent.

Embodiment 28

The grid-tied converter according to any of embodiments 25-27, furthercomprising an indirect controller coupled to the input current andproviding an output current v_(ac-CHB2) to the SHCM-PWM unit and thePSPWM unit.

Embodiment 29

The grid-tied converter according to any of embodiments 25-28, furthercomprising a phase lock loop (PLL) coupled to the modulation selectorand an output of the grid-tied converter.

Embodiment 30

The grid-tied converter according to any of embodiments 25-29, furthercomprising an inductor connected to the H bridge converter.

Embodiment 31

A motor, comprising:

a selective harmonic current mitigation pulse width modulation(SHCM-PWM) unit coupled to an input current and providing an outputsignal SW_(SHCM);

a phase shift pulse width modulation (PSPWM) unit coupled to the inputcurrent and providing an output signal SW_(PS);

a modulation selector coupled to the output signal SW_(SHCM) of theSHCM-PWM unit and the output signal SW_(PS) of the PSPWM unit andproviding an output signal SW;

a H bridge converter coupled to the output signal SW of the modulationselector; and

a motor connected to the H bridge converter.

Embodiment 32

The motor according to embodiment 31, wherein the motor is a singlephase asynchronous motor or a three phase asynchronous motor.

Embodiment 33

A filter, comprising:

a selective harmonic current mitigation pulse width modulation(SHCM-PWM) unit coupled to an input current and providing an outputsignal SW_(SHCM);

a phase shift pulse width modulation (PSPWM) unit coupled to the inputcurrent and providing an output signal SW_(PS);

a modulation selector coupled to the output signal SW_(SHCM) of theSHCM-PWM unit and the output signal SW_(PS) of the PSPWM unit andproviding an output signal SW;

a H bridge converter coupled to the output signal SW of the modulationselector; and

a passive filter connected to the H bridge converter.

Embodiment 34

The filter according to embodiment 33, wherein the passive filterincludes at least one of an L filter, an LC filter, and an LCL filter.

A greater understanding of the present invention and of its manyadvantages may be had from the following examples, given by way ofillustration. The following examples are illustrative of some of themethods, applications, embodiments, and variants of the presentinvention. They are, of course, not to be considered as limiting theinvention. Numerous changes and modifications can be made with respectto the invention.

EXAMPLE 1

For performance evaluation of the hybrid SHCM-PWM and PSPWM technique,MATLAB Simulink was used for the simulations. The circuit parameters,which were used in both simulation and experimental results, are shownin Table II. The DC voltage of battery for each cell in the simulationand experimental results was 65V. The obtained solutions in FIG. 4 canstill be used for V_(dc)=65 V because low DC link voltages result in lowvoltage harmonics in equations (2) and (3).

The purposes of the simulations and experiments were to: (a) validatewhether i_(in) can meet the IEEE 519 current harmonic limits with theextended solution range in equation (6); (b) validate whether the hybridSHCM-PWM and PSPWM technique based on FIG. 6 can achieve high dynamicresponse and the transient current can be significantly reduced; and (c)validate whether the CHB converter can process four-quadrant active andreactive power. The active and reactive powers can be either injected toor absorbed from the power grid.

FIGS. 10(a)-10(d) show first simulation results for a conventionalfour-quadrant converter and a four-quadrant converter according to anembodiment of the subject invention. In the first comparativesimulations, the active and reactive power flowing from power grid tothe converter changed from 1000 W−1000VAR to 200 W+250VAR at t=0.60231seconds (s) and then changed back to 1000 W−1000VAR at t=0.6667 s. Inthe first simulation in FIG. 10(a) for a conventional SHCM-PWMtechnique, during the transient condition, when the active and reactivepowers are changed, more than two fundamental cycles are required toreach steady state. The DC offset of the current i_(in) lasts for morethan two cycles. The maximum DC offset shown in Table III is 43%-78%.This DC offset can lead to instability. The waveforms in FIG. 10(b) arefor a hybrid technique of an embodiment of the subject invention. Duringthe transient condition, the active power and reactive power are changedseparately within one cycle with the conditions defined in equation(17). Referring to FIG. 10(b), i_(in) reaches the steady state withinless than one cycle. The maximum 1.75%-8.5% DC offset is negligible asshown in Table III. The carrier frequency of the PSPWM technique underdynamic condition is 240 Hz as derived in equation (20). The harmonicspectrum of i_(in) with the embodiment, when the active and reactivepower is 1000 W-1000VAR, is shown in FIG. 10(c). The modulation index ofv_(ac-CHB) is 2.399, which is within the modulation index range of theconventional technique in FIG. 4. The harmonic spectrum of i_(in) withthe embodiment of the subject invention, when the active and reactivepower is 200 W+250VAR, is shown in FIG. 10(d). The modulation index is1.647, which is inside the extended modulation index range in FIG. 4. Asshown in FIGS. 10(c) and 10(d), with the embodiment of the subjectinvention, the harmonic spectra of i_(in) can meet IEEE 519 currentharmonic limits.

TABLE III THE MAXIMUM DC OFFSET OF I_(IN) IN SIMULATIONS WITH EITHERCONVENTIONAL OR PROPOSED TECHNIQUES 1st 2nd Comparative 1st transitiontransition 2nd transition transition Simulations (conventional)(proposed) (conventional) (proposed) 1 78% 8.5% 43% 1.75% 2 92% 3.8% 94% 1.4%

FIGS. 11(a)-11(d) show second simulation results for a conventionalfour-quadrant converter and a four-quadrant converter according to anembodiment of the subject invention. In the second comparativesimulations, the active and reactive power flowing from power grid tothe converter change from −850 W+825VAR to −500 W−600VAR at t=0.6023 sand then change back to −850 W+825VAR at t=0.6667 s. In FIG. 11(a), forthe conventional SHCM-PWM technique, during the transient condition,when the active and reactive powers are changed, more than twofundamental cycles are required to reach steady state. A huge 92%-94% DCoffset shown in Table III is observed in the current i_(in), and itlasts for more than two cycles. This DC offset can lead to instabilityon the controller. FIG. 11(b) shows a second simulation result ofV_(ac-CHB), V_(ac-Grid), and I_(in) for a four-quadrant converteraccording to an embodiment of the subject invention. During thetransient condition, the active power and reactive power are changedseparately within one cycle with the conditions defined in equation(17). Referring to FIG. 11(b), i_(in) reaches the steady state withinless than one cycle. The maximum 1.4%-3.8% DC offset is negligible asshown in Table III. The carrier frequency of the PSPWM technique underdynamic condition is 240 Hz as derived in equation (20). The harmonicspectrum of i_(in) at −850 W+825VAR with a 1.52 modulation index, whichis inside the extended modulation index range in FIG. 4, is shown inFIG. 11(c). The harmonic spectrum of i_(in) at −500 W−600VAR is shown inFIG. 11(d). The modulation index is 2.274, which is inside theconventional modulation index range in FIG. 4. It is apparent that theharmonics of i_(in) for both conditions meet the current harmonic limitsof IEEE 519.

EXAMPLE 2

A 7-level four-quadrant CHB converter according to an embodiment of thesubject invention, having the same parameters as in the simulations, wasfabricated and investigated. FIG. 12 shows a hardware prototype of sucha four-quadrant CHB. The TMS320F28335 DSP was used in the prototype.Similar to FIGS. 10(a)-10(d), in the first comparative experiments, theactive and reactive power flowing from power grid to the converterchanged from 1000 W−1000VAR to 200 W+250VAR and then changed back to1000 W−1000VAR. The transient periods are between the two red (vertical)lines in FIGS. 13(a), 13(b), 14(a), and 14(b).

FIGS. 13(a)-13(d) show first experimental results for a conventionalfour-quadrant converter and a four-quadrant converter according to anembodiment of the subject invention. In FIG. 13(a), the conventionaltechnique with SHCM-PWM takes at least two fundamental cycles to reachthe steady state. A 35%-65% DC offset is observed in i_(in) in Table IVduring the transient. This large DC offset can lead to instability ofthe controller and can reduce the reliability of the semiconductorswitches. FIG. 13(b) shows a first experimental result of V_(ac-CHB),V_(ac-Grid), and I_(in) for a four-quadrant converter according to anembodiment of the subject invention. In FIG. 13(b), for the hybridtechnique of an embodiment of the subject invention, which complies withthe condition derived in equation (17), each of the d and q componentschanges once in one cycle. It takes less than one cycle to reach steadystate. The 2.5%-12% DC offset is much smaller than that of theconventional technique as shown in Table IV. The switching frequency ofthe PSPWM technique is 240 Hz as derived in equation (20). The currentharmonic spectra of i_(in) in steady state at both 1000 W−1000VAR and200 W+250VAR are shown in FIGS. 13(c) and 13(d), respectively. Bothharmonic spectra can meet the IEEE 519 current harmonic limits. Themodulation indices for both conditions are the same as the simulationresults. This confirms the extended modulation index range in FIG. 4.

TABLE IV THE MAXIMUM DC OFFSET OF I_(IN) IN EXPERIMENTS WITH EITHERCONVENTIONAL OR PROPOSED TECHNIQUES First Second Second Experiment Firsttransition transition transition transition number (conventional)(proposed) (conventional) (proposed) First 65% 12% 35% 2.5% Second 60% 5% 37% 5.8%

FIGS. 14(a)-14(d) show second experimental results for a conventionalfour-quadrant converter and a four-quadrant converter according to anembodiment of the subject invention. In the second comparativeexperiments, the active and reactive power flowing from power grid tothe converter change from −850 W+825VAR to −500 W−600VAR and then changeback to −850 W+825VAR.

Referring to FIG. 14 (a), the conventional technique takes at least twofundamental cycles to achieve steady state. A 37%-60% DC offset isobserved during the transient in Table IV. In FIG. 14 (b), the hybridtechnique of the embodiment of the subject invention takes less than onecycle to reach steady state with only a 5%-5.8% DC offset during thetransient condition. FIGS. 14(c) and 14(d) show that the currentharmonic spectra of izn meet IEEE 519 current harmonic limits under bothconditions.

As demonstrated by both experimental and simulation results, with thehybrid techniques of embodiments of the subject invention, the CHBrectifier can process four-quadrant active and reactive power with theextended modulation index, achieve fast dynamic response, and meetIEEE-519 current harmonic limits.

The hybrid techniques of embodiments of the subject invention canachieve a transient free dynamic response because of the non-idealcomponent parameters, such as the variations of the DC link voltages,the resistance of the inductor, and the impedance of power grid (as wellas possibly others), while maintaining a small DC offset during thetransient condition. In addition, compared with conventional techniques,the hybrid techniques of embodiments of the subject inventionsignificantly improve the dynamic response.

EXAMPLE 3

FIGS. 15(a)-15(c) show another example according to an embodiment of thesubject invention. FIG. 15(a) shows a grid-tied converter thatselectively uses a selective harmonic current mitigation pulse widthmodulation (SHCM-PWM) unit and a phase shift pulse width modulation(PSPWM) unit. Referring to FIGS. 6 and 15(a), the grid-tied converter ofFIG. 15(a), indicated as colored H bridges, replaces the position of theCHB converter of FIG. 6, thereby providing high efficiency and highdynamic performance. The grid-tied converter further includes inductorsL_(A), L_(B), and L_(C) connected to a grid such as Neutral PointClamped (NPC) and Flying Capacitor (FC).

FIG. 15(b) shows a three-phase asynchronous motor according to anembodiment of the subject invention. Similar to FIG. 15(a), thethree-phase asynchronous motor replaces the position of the CHBconverter of FIG. 6, thereby improving the dynamic performance in driveapplication using the three-phase asynchronous motor. The motor can be asingle phase asynchronous motor.

FIG. 15(c) shows a filter according to an embodiment of the subjectinvention. Similar to FIGS. 15(a) and 15(b), the filter replaces theposition of the CHB converter of FIG. 6, thereby improving dynamicperformance in any kind of passive filter. The filter includes any kindof passive filters including an L filter, an LC filter, and an LCLfilter.

In addition to above described examples, a hybrid PS-PWM and asymmetricSHCM-PWM technique is developed to reduce the switching frequency andinductance of the PS-PWM technique and meet the limits of IEEE 519 2014.To reach this goal, the voltage harmonics due to the PS-PWM techniqueare mitigated with the harmonics generated from the low-frequencyasymmetric SHCM-PWM technique. Consequently, the switching frequency isreduced. Moreover, a general equation set for the hybrid PS-PWM andasymmetric SHCM-PWM technique is derived based on the equations of thePS-PWM and asymmetric SHCM-PWM techniques.

Guidelines are developed for the design of critical parameters such asthe coupling inductance and the switching frequency of the hybridmodulation technique. The hybrid modulation technique increases powerefficiency, reduces inductance, meets the limits of IEEE 519, and canachieve four-quadrant operation for grid-tied CHB converters.

Moreover, the best and worst scenarios for changing the active andreactive current of the grid-tied converter are derived. So, instead ofchanging the active and reactive current twice in a fundamental cycle asdiscussed, the active and reactive current can be changed just once ineach half-period. Also, the effects of low-order current harmonics ofthe grid-tied converter on the DC transient response will be discussedand the conditions which can cause the best and worst scenarios for theDC transient response will be derived. Using high-switching frequencymodulation techniques such as PS-PWM can achieve a high-dynamicperformance due to eliminating the low-order harmonics and simplicity ofcontrolling the fundamental and low-order harmonics of the CHB. Also,using the asymmetric SHCM-PWM can increase the efficiency of theconverter. So the hybrid technique can be used for different types ofgrid-tied converters such as electric vehicle charging stations andsmart grids to have both advantages.

Hybrid PS-PWM and Asymmetric SHCM-PWM Technique

PS-PWM Technique

The configuration of a power grid-tied CHB converter is shown in FIG.16, where N plus P H-bridge cells are cascaded to generate the CHBvoltage v_(ac-CHB)(t) in equation (21) below to control the AC currenti_(in)(t) injected to the power grid v_(ac-Grid)(t). Here, the N cellsof the CHB grid-tied converter is modulated by the PS-PWM and P cellsare modulated by the asymmetric SHCM-PWM technique. The DC link voltageof each cell is V_(dc). L is the coupling inductance betweenv_(ac-CHB)(t) and v_(ac-Grid)(t). In this technique, the CHB converterincludes i number of cells, i=N+P, they are modulated with the PS-PWM toachieve better dynamic performance than low-frequency modulationtechniques. P cells are modulated with the asymmetric SHCM-PWM to meetcurrent harmonic limits. It is derived the magnitudes and phases for theharmonic voltages of N CHB cells with the PS-PWM.

$\begin{matrix}{{v_{{ac} - {CHB} - {PSPWM}}(t)} = {{{NV}_{dc}M\;{\cos\left( {{\omega_{o}t} + \theta_{o}} \right)}} + {\frac{4V_{dc}}{\pi}{\prod\limits_{B = 1}^{\infty}\;{\sum\limits_{A = {- \infty}}^{\infty}\;{\frac{1}{2B}{J_{{2A} - 1}\left( {{NB}\;\pi\; M} \right)} \times {\sin\left( {\left( {{2{NB}} + {2A} - 1} \right)\frac{\pi}{2}} \right)}{\cos\left( {{2{NB}\;\omega_{c}t} + {\left( {{2A} - 1} \right)\left( {{\omega_{o}t} + \theta_{o}} \right)}} \right)}}}}}}} & (21)\end{matrix}$where B and A are the baseband and sideband harmonic orders,respectively. v_(ac-CHB-PSPWM)(t) is the voltage of the N-cell CHB withthe PS-PWM technique, J is the Bessel function of the first kind.ω_(c)=2πf_(c) is the carrier frequency (radian) and f_(c) is the carrierfrequency (Hz). ω_(o)=2πf_(o), θ_(o), f_(o) are the frequency (radian),phase and frequency (Hz) of the fundamental of v_(ac-CHB-PSPWM)(t). Thetotal modulation index of the CHB converter is M_(a)=M(N+P), where M isthe average modulation index of all cells. The DC links of these CHBcells can be connected to DC/DC converters.

The harmonic magnitudes of (21) are shown in FIG. 17 where f_(s) is theswitching frequency of v_(ac-CHB-PSPWM)(t) in each half-period andf_(s)=2Nf_(c). By expanding sin((2NB+2 A−1)π/2) in (21), the followingequation (22) can be derived,

$\begin{matrix}{{v_{{ac} - {CHB} - {PSPWM}}(t)} = {{{NV}_{dc}M\;{\cos\left( {{\omega_{o}t} + \theta_{o}} \right)}} + {\frac{4V_{dc}}{\pi}{\prod\limits_{B = 1}^{\infty}\;{\sum\limits_{A = {- \infty}}^{\infty}\;{\frac{1}{2B}{J_{{2A} - 1}\left( {{NB}\;\pi\; M} \right)}{\cos\left( {{NB}\;\pi} \right)} \times {\sin\left( {\left( {{2A} - 1} \right)\frac{\pi}{2}} \right)}{\cos\left( {{2{NB}\;\omega_{c}t} + {\left( {{2A} - 1} \right)\left( {{\omega_{o}t} + \theta_{o}} \right)}} \right)}}}}}}} & (22)\end{matrix}$

By shifting fundamental by −90° and harmonics by −(2 A−1) 90°, thesecond term in (22) can be further written as,

$\begin{matrix}{{v_{{ac} - {CHB} - {PSPWM}}(t)} = {{{NV}_{dc}M\;{\sin\left( {{\omega_{o}t} + \theta_{o}} \right)}} + {\frac{4V_{dc}}{\pi}{\prod\limits_{B = 1}^{\infty}\;{\sum\limits_{A = {- \infty}}^{\infty}\;{\frac{1}{2B}{J_{{2A} - 1}\left( {{NB}\;\pi\; M} \right)}{\cos\left( {{NB}\;\pi} \right)}{\sin\left( {\left( {{2A} - 1} \right)\frac{\pi}{2}} \right)}{\cos\left( {{2{NB}\;\omega_{c}t} + {\left( {{2A} - 1} \right)\left( {{\omega_{o}t} + \theta_{o} - \frac{\pi}{2}} \right)}} \right)}}}}}}} & (23)\end{matrix}$

The second term in (23) consists of the harmonic components of the CHBvoltage. It can be decomposed into the sine and cosine components as;

$\begin{matrix}{{{v_{{ac} - {CHB} - {PSPWM}}(t)} = {{{NV}_{dc}M\;{\sin\left( {{\omega_{o}t} + \theta_{o}} \right)}} + {\frac{4V_{dc}}{\pi}{\sum\limits_{B = 1}^{\infty}\;{\sum\limits_{A = {- \infty}}^{\infty}\;{\frac{1}{2B}{J_{{2A} - 1}\left( {{NB}\;\pi\; M} \right)}{\cos\left( {{NB}\;\pi} \right)}{\sin\left( {\left( {{2A} - 1} \right)\theta_{o}} \right)}{\cos\left( {{2{NB}\;\omega_{c}t} + {\left( {{2A} - 1} \right)\left( {\omega_{o}t} \right)}} \right)}}}}} + {\frac{4V_{dc}}{\pi}{\sum\limits_{B = 1}^{\infty}\;{\sum\limits_{A = {- \infty}}^{\infty}\;{\frac{1}{2B}{J_{{2A} - 1}\left( {{NB}\;\pi\; M} \right)}{\cos\left( {{NM}\;\pi} \right)}{\cos\left( {\left( {{2A} - 1} \right)\theta_{o}} \right)}{\sin\left( {{2{NB}\;\omega_{c}t} + {\left( {{2A} - 1} \right)\left( {\omega_{o}t} \right)}} \right)}}}}}}},} & (24)\end{matrix}$

As shown in (24), the magnitudes of the sine and cosine componentsdepend on the phase θ₀ of the fundamental. As a result, the magnitudesof the harmonics of the PS-PWM cannot be controlled if the fundamentalis controlled. The asymmetric SHCM-PWM technique can control both themagnitude and phase of each harmonic, so it is employed in the hybridmodulation technique to mitigate the harmonics to meet IEEE 519. Themagnitudes of the harmonics in (24) when B is less than or equal to 2are shown in FIG. 17, which shows the magnitude versus bandwidth. Thebandwidth BW_(B) of the B^(th) order baseband in FIG. 17 is given byCarlson's rule:BW _(B)≈2(NMBπ+2)f ₀  (25)

In FIG. 17, the overlap of BW₁ and f₀ should be avoided becauseswitching harmonics will influence the fundamental. At the same time,f_(s) which is equal to 2Nf_(c), should be as low as possible tominimize switching power loss. To achieve these two goals, the followingcondition should be met,f _(s) −BW ₁ >f ₀ ⇒f _(s) >f ₀ +BW ₁  (26)Asymmetric SHCM-PWM Technique

For P cells that use the asymmetric SHCM-PWM in FIGS. 16 and 18, if thefundamental phase of the P-cell CHB voltage v_(ac-CHB-ASHCM)(t) is equalto θ_(o), its Fourier series is,

$\begin{matrix}{{{v_{{ac} - {CHB} - {ASHCM}}(t)} = {\sum\limits_{h = 1}^{\infty}\;\left( {{\frac{2V_{dc}}{\pi\; h}\left( {{- {\sin\left( {h\;\theta_{1}} \right)}} + {\sin\left( {h\;\theta_{2}} \right)} - \ldots + {\sin\left( {h\;\theta_{K}} \right)}} \right){\cos\left( {{h\;\omega_{o}t} + {h\;\theta_{o}}} \right)}} + {\frac{2V_{dc}}{\pi\; h}\left( {{\cos\left( {h\;\theta_{1}} \right)} - {\cos\left( {h\;\theta_{2}} \right)} + \ldots - {\cos\left( {h\;\theta_{K}} \right)}} \right){\sin\left( {{h\;\omega_{o}t} + {h\;\theta_{o}}} \right)}}} \right)}},} & (27)\end{matrix}$where h is the harmonic order, K is the number of switching transitionsin each half-period for the P-cell CHB. Because of the asymmetricwaveform generated by the asymmetric SHCM-PWM in FIG. 3 has a half-wavesymmetry, the even order harmonics in (7) are equal to zero. θ₁, θ₂, . .. , and θ_(K) are switching angles representing the switchingtransitions of the asymmetric SHCM-PWM in each half-period. By expandingcos(hω_(o)t+hθ_(o)) and sin(hω_(o)t+hθ_(o)) in (27) and usingmathematical manipulations, (27) can be rewritten as,

$\begin{matrix}{{{v_{{ac} - {CHB} - {ASHCM}}(t)} = {\sum\limits_{h = 1}^{\infty}\;\left( {{\frac{2V_{dc}}{\pi\; h}\left( {{- {\sin\left( {{h\;\theta_{1}} - {h\;\theta_{o}}} \right)}} + {\sin\left( {{h\;\theta_{2}} - {h\;\theta_{o}}} \right)} - \ldots + {\sin\left( {{h\;\theta_{K}} - {h\;\theta_{o}}} \right)}} \right){\cos\left( {h\;\omega_{o}t} \right)}} + {\frac{2V_{dc}}{\pi\; h}\left( {{\cos\left( {{h\;\theta_{1}} - {h\;\theta_{o}}} \right)} - {\cos\left( {{h\;\theta_{2}} - {h\;\theta_{o}}} \right)} + \ldots - {\cos\left( {{h\;\theta_{K}} - {h\;\theta_{o}}} \right)}} \right){\sin\left( {h\;\omega_{o}t} \right)}}} \right)}},} & (28)\end{matrix}$

In (28), the sine and cosine terms of the harmonics are decomposed. Bycontrolling the switching angles θ₁, θ₂, and θ_(K), the low-orderharmonics due to the PS-PWM in N-cell CHBs can be compensated with theharmonics generated by the asymmetric SHCM-PWM in (28).

Derivation of Equations for Hybrid PS-PWM and Asymmetric SHCM-PWMTechnique

In FIG. 16, the h^(th) order current harmonic I_(in-h) can be calculatedas,

$\begin{matrix}{{{I_{{in} - h}} = {\frac{\begin{matrix}{{V_{{ac} - {CHB} - {PSPWM} - h}\angle\;\theta_{{PSPWM} - h}} +} \\{V_{{ac} - {CHB} - {ASHCM} - h}\angle\;\theta_{{ASHCM} - h}}\end{matrix}}{j\;\omega_{o}{hL}}}},} & (29)\end{matrix}$

where V_(ac-CHB-PSPWM-h), V_(ac-CHB-PSPWM-h), θ_(PSPWM-h) andθ_(ASHCM-h) are the magnitudes and phases of the h^(th) order voltageharmonic due to the PS-PWM and asymmetric SHCM-PWM respectively.I_(in-h) must meet the total demand distortion (TDD) and harmonic limitsup to 50^(th) order specified by IEEE 519 2014 in Table I. I_(L) is themaximum demand load current and I_(sc) is the short circuit current atthe PCC. To meet the current limits of IEEE 519, the following equationmust be met,

$\begin{matrix}{{{\frac{I_{{in} - h}}{I_{L}}} = {{\frac{\begin{matrix}{{V_{{ac} - {CHB} - {PSPWM} - h}\angle\;\theta_{{PSPWM} - h}} +} \\{V_{{ac} - {CHB} - {ASHCM} - h}\angle\;\theta_{{ASHCM} - h}}\end{matrix}}{j\;\omega_{o}{hLI}_{L}}} \leq C_{h}}},} & (30)\end{matrix}$where C_(h) is the limit of the h^(th) order current harmonic in Table Iin the worst scenario based on the short circuit ratio. Based on IEEE5192014, TDD must also meet the standard below,

$\begin{matrix}{{T_{DD} = {\sqrt{\left( \frac{I_{{in} - 3}}{I_{L}} \right)^{2} + \left( \frac{I_{{in} - 5}}{I_{L}} \right)^{2} + \ldots + \left( \frac{I_{{in} - 49}}{I_{L}} \right)^{2}} \leq C_{TDD}}},} & (31)\end{matrix}$where C_(TDD) is the TDD limit of IEEE 519. It is worth to mention thatthe grid voltage harmonic requirements of IEEE-519 is not consideredhere.

TABLE I HARMONIC LIMITS OF IEEE 519 I_(sc)/I_(L) ≤ 20 <11 11 ≤ h < 17 17≤ h < 23 23 ≤ h < 35 35 ≤ h < 50 C_(TDD) C_(h & TDD) 4% 2% 1.5% 0.6%0.3% 5%

Based on (24), (28), (30) and (31), the equation set which is going tobe used to calculate switching angles using optimization techniques forthe hybrid modulation technique is therefore becoming,

$\begin{matrix}\left\{ \begin{matrix}{{V_{{ac} - {CHB} - 1} = {\frac{1}{\sqrt{2}}\sqrt{\begin{matrix}\left( {\frac{2V_{dc}}{\pi}\left( {{\cos\left( {\theta_{1} - \theta_{o}} \right)} - {\cos\left( {\theta_{2} - \theta_{o}} \right)} + \ldots -} \right.} \right. \\{\left. {\left. {\cos\left( {\theta_{k} - \theta_{o}} \right)} \right) + {{NMV}_{dc}{\cos\left( \theta_{o} \right)}}} \right)^{2} +} \\\left( {\frac{2V_{dc}}{\pi}\left( {{- {\sin\left( {\theta_{1} - \theta_{o}} \right)}} + {\sin\left( {\theta_{2} - \theta_{o}} \right)} - \ldots +} \right.} \right. \\\left. {\left. {\sin\left( {\theta_{k} - \theta_{o}} \right)} \right) + {{NMV}_{dc}{\sin\left( \theta_{o} \right)}}} \right)^{2}\end{matrix}}}},} \\{{\frac{\sqrt{\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\left( {\frac{4V_{dc}}{\pi}{\sum\limits_{B = 1}^{\infty}\;\left( {\sum\limits_{A = {- \infty}}^{\infty}\;\left( {\frac{1}{2B}{J_{{2A} - 1}\left( {{NB}\;\pi\; M}\; \right)}{\cos\left( {{NB}\;\pi} \right)}} \right.} \right.}} \right. \\{\left. \left. {\sin\left( {\left( {{2A} - 1} \right)\theta_{o}} \right)} \right) \right) + {\frac{2V_{dc}}{\pi\; h}\left( {{- {\sin\left( {h\left( {\theta_{1} - \theta_{o}} \right)} \right)}} +} \right.}}\end{matrix} \\{\left. \left. {{\sin\left( {h\left( {\theta_{2} = \theta_{o}} \right)} \right)} - \ldots + {\sin\left( {h\left( {\theta_{k} - \theta_{o}} \right)} \right)}} \right) \right)^{2} +}\end{matrix} \\\left( {\frac{4V_{dc}}{\pi}{\sum\limits_{B = 1}^{\infty}\;\left( {\sum\limits_{A = {- \infty}}^{\infty}\;\left( {\frac{1}{2B}{J_{{2A} - 1}\left( {{NB}\;\pi\; M} \right)}{\cos\left( {{NB}\;\pi} \right)}} \right.} \right.}} \right.\end{matrix} \\{\left. \left. \left. {{\cos\left( {{2A} - 1} \right)}\theta_{o}} \right) \right) \right) + {\frac{2V_{dc}}{\pi\; h}\left( {{\cos\left( {h\left( {\theta_{1} - \theta_{o}} \right)} \right)} -} \right.}}\end{matrix} \\\left. \left. {{\cos\left( {h\left( {\theta_{2} - \theta_{o}} \right)} \right)} + \ldots - {\cos\left( {h\left( {\theta_{k} - \theta_{0}} \right)} \right)}} \right) \right)^{2}\end{matrix}}}{{\omega_{o}{hLI}_{L}}} \leq C_{h}},{h = 3},5,7,\ldots\mspace{14mu},} \\{\sqrt{\left( \frac{I_{{in} - 3}}{I_{L}} \right)^{2} + \left( \frac{I_{{in} - 5}}{I_{L}} \right)^{2} + \ldots + \left( \frac{I_{{in} - 49}}{I_{L}} \right)^{2}} \leq C_{TDD}}\end{matrix} \right. & (32)\end{matrix}$

In (32), the first equation (31) is used to generate the desiredfundamental CHB voltage. The second and the third equations areconditions to meet the limits of current harmonics and TDD,respectively. For the sideband harmonics outside the bandwidth definedin (25), because their magnitudes are small, they will be ignored. Onlythe sideband harmonics within the bandwidth will be considered in (32).Therefore, the number of sideband harmonics of each baseband harmonic in(32) is a function of its bandwidth (25). The switching angles in (32)can be solved using optimization technique such as particle swarmoptimization technique, if critical parameters such as V_(dc), I_(L), K,N, P, and L are determined first.

Parameter Design for the Hybrid Modulation Technique

In (32), V_(dc), I_(L), K, N, P, and L must be designed for the hybridmodulation technique. To achieve reactive power compensation, themagnitude of the fundamental CHB voltage should be higher than theamplitude of the grid voltage V_(ac-Grid-1), so

$\begin{matrix}\left. {\frac{4{V_{dc}\left( {N + P} \right)}}{\pi} \geq {\sqrt{2}V_{{ac} - {Grid} - 1}}}\Rightarrow{V_{dc} \geq \frac{\sqrt{2}\pi\; V_{{ac} - {Grid} - 1}}{4\left( {N + P} \right)}} \right. & (33)\end{matrix}$

I_(L) is determined based on the maximum volt-ampere of the grid-tiedCHB converter. I_(L) is 14.14 A, V_(ac-Grid-1) is 110V and volt-ampereis 1.555 kVA. Because N cells use the PS-PWM and P cells use theasymmetric SHCM-PWM, there is a trade-off between choosing N and P indifferent applications. If N≥P, the converter has better a dynamicperformance as will be discussed later than N≤P [20], however, theswitching power loss is higher than N≤P, and the inductance which canmitigate the current harmonics is bigger than N≤P. If N≤P, the convertercan mitigate more number of current harmonics with lower switching lossthan N≥P while the dynamic performance is not as good as N≥P. Theobjective is to use a 3-cell CHB converter to demonstrate that thehybrid modulation technique is better than the conventional PS-PWMtechnique in meeting harmonic limits with smaller switching frequencyand L, so N=1 and P=2. In (25), because M=1, N=1 and B=1,BW_(B)=10.28f₀. Based on (6), f_(s)>677 Hz, so f_(s) is selected as 720Hz. The switching transitions in each half-period is therefore 12.

To solve inductance and switching angles from equation set (32) for thedesired fundamental and meeting both the current harmonic and TDD limitsof IEEE 519, the range of K and L should be determined. In (30), in theworst scenario, the harmonic mitigation due to v_(ac-CHB-ASHCM-h) isignored, so the worst inductance L_(worst) is given by (34). Because ofthe harmonic mitigation due to v_(ac-CHB-ASHCM-h), the actual inductanceis smaller than L_(worst) as shown in (34).

$\begin{matrix}{{{L \leq L_{worst}} = {\max\left( \frac{V_{{ac} - {CHB} - {PSPWM} - h}}{\omega_{o}{hI}_{L}C_{h}} \right)}},{{{for}\mspace{14mu} h} = 3},5,7,\ldots} & (34)\end{matrix}$

Also, to meet TDD limit without the contribution from the asymmetricSHCM-PWM, based on (31), inductance L should meet,

$\begin{matrix}{L \geq {\frac{1}{C_{TDD}}\sqrt{\sum\limits_{h = 3}^{49}\;\left( \frac{\max\left( V_{{ac} - {CHB} - {PSPWM} - h} \right)}{2\;\pi\;{hf}_{o}I_{L}} \right)^{2}}}} & (35)\end{matrix}$

(34) and (35) give the range of the inductance L when solving equationset (32). In equation set (32), due to the fact that significantharmonics, which could be over the limit of the PS-PWM are within thesidebands of carrier fundamental and harmonics, the low-order voltageharmonics: 3rd, 5th, . . . , ((f_(s)−BW₁−60)/60)^(th), of the PS-PWM arevery low in FIG. 17. The asymmetric SHCM-PWM can mitigate thesignificant harmonics generated by the PS-PWM within the sidebands ofcarrier fundamental and harmonics.

FIG. 18 shows a time domain waveform of the asymmetric SHCM-PWM within ahalf-period, according to an embodiment of the subject invention.

FIG. 19 shows the calculated highest inductance to meet harmonic limitsfor each order of harmonics based on (24) and (34). The maximuminductance 0.766 pu happened at 11^(th) order harmonic.

FIG. 20 shows the calculated lowest inductances based on (34) and (35)at different TDDs. For IEEE 519, the inductance is 0.363 pu.

FIGS. 21(a) and 21(b) show phase diagrams of h^(th) order harmonics. Thephasor diagram of (29) and (30) is shown in FIG. 21 (a) for the h^(th)order harmonic when |v_(ac-CHB-PSPWM-h)|≥|jω_(o)hC_(h)LI_(L)|.θ_(PSPWM-h) is from 0 to 2π. To meet the harmonic limit C_(h) in (30),the head of vector V_(ac-CHB-PSPWM-h) should always be located withinthe circle, which centers at the end of vector V_(ac-CHB-PSPWM-h), withradius |jω_(o)hC_(h)LI_(L)|. In FIG. 21 (b), when|V_(ac-CHB-PSPWM-h)|<|jω_(o)hC_(h)LI_(L)|, the magnitude of theV_(ac-CHB-ASHCM-h), no matter what phase it has, only needs to be lessthan |jω_(o)hC_(h)LI_(L)|−|V_(ac-CHB-PSPWM-h)| to meet the currentharmonic limit. It is relatively easy to meet the harmonic limit forthis case and it has been discussed.

The range of V_(ac-CHB-ASHCM-h), in which I_(in-h) can meet the harmoniclimit, when |V_(ac-CHB-PSPWM-h)|≥|jω_(o)hC_(h)LI_(L)|, is given by thered circle which centers at the origin with the radius equal tomax(|V_(ac-CHB-PSPWM-h)|+|jω_(o)hC_(h)LI_(L)|). The range is given by(36).|V _(ac)-CHB-ASHCM-h|≤max(|ω_(o) hLI _(L) C _(k) |+|V_(ac-CHB-PSPWM-h)|)  (36)

At N=1, when the modulation index is within the PS-PWM's normalmodulation index range [0, 0.785], the maximum magnitudes of the voltageharmonics of the PS-PWM in the normal modulation index range can becalculated from (34) and are shown in FIG. 22. In FIG. 22,|V_(ac-CHB-PSPWM-h)|≥|jω_(o)hC_(h)LI_(L)| at the 11th, 13th, 23rd, 25th,27th, 35th, 37th, and 39th harmonics. So, for these orders, (36) shouldbe used to design the asymmetric SHCM-PWM. Also, the maximum harmonicmagnitude |v_(ac-CHB-ASHCM-h)|_(max) of the asymmetric SHCM-PWM is,

$\begin{matrix}{{v_{{ac} - {CHB} - {ASHCM} - h}}_{\max} = \frac{2V_{dc}K}{\pi\; h}} & (37)\end{matrix}$

Based on (36) and (37), when |v_(ac-CHB-PSPWM-h)|>|jω_(o)hC_(h)LI_(L)|,for h^(th) harmonic, the switching transitions K of the asymmetricSHCM-PWM in each half-period should meet the condition below,

$\begin{matrix}{K \leq {\frac{\pi\; h}{2V_{dc}}\left( {\max\left( {{{\omega_{o}{hLI}_{L}C_{h}}} + {V_{{ac} - {CHB} - {PSPWM} - h}}} \right)} \right)}} & (38)\end{matrix}$(38) gives a maximum initial K to solve equation set (32).

FIG. 23 shows the calculated maximum switching transitions in ahalf-period for a P-cell asymmetric SHCM-PWM based on (38) for differentharmonic orders. The 37^(th) harmonic requires the highest number ofcalculated transitions (13.4). Based on (38) and FIG. 23, because 12 isthe largest even number below 13.4, the initial K of the asymmetricSHCM-PWM to solve (32) should be 12. K should always be an even numberand can be further reduced as long as (32) has solutions during theoptimization process.

After finding all of these important parameters for (32), forfour-quadrant active and reactive power operation in FIG. 1, the desiredmodulation index M_(a) and the initial phase θ_(o) of the CHB voltageare

$\begin{matrix}{M_{a} = {\frac{{\sqrt{2}V_{{ac} - {Grid} - 1}{\angle 0}} - {\left( {R + {j\;\omega_{o}L}} \right)\sqrt{2}I_{{in} - 1}{\angle\theta}_{in}}}{4\;{V_{dc}/\pi}}}} & (39) \\{\theta_{o} = {\arg\left( \frac{{\sqrt{2}V_{{ac} - {Grid} - 1}{\angle 0}} - {\left( {R + {j\;\omega_{o}L}} \right)\sqrt{2}I_{{in} - 1}{\angle\theta}_{in}}}{4\;{V_{dc}/\pi}} \right)}} & (40)\end{matrix}$where θ_(m) is the desired phase of the fundamental AC current.

FIG. 24 shows the calculated phases and modulation indices based on (39)and (40) for the four-quadrant active and reactive power operationPS-PWM and asymmetric SHCM-PWM technique when the magnitude of the ACcurrent changes between 0 A and 20 A and the phase of the AC currentchanges between 0° and 360°.

Due to the difficulties of showing all of the switching angle solutionsfor different current magnitudes and phases, FIG. 25 only shows samplesolutions for 41 operating points of the hybrid PS-PWM and asymmetricSHCM-PWM technique, when the phase θ_(in) of the fundamental current isin phase or out of phase with the phase θ_(grid) of the fundamental gridvoltage and the current magnitude i_(in-1)(t) varies between 0 A and 20A and is in phase or out of phase with v_(ac-Grid-1)(t). The actualsolutions of the technique are in a wide modulation index range as inFIG. 24.

Dynamic Response Analysis of the Grid-Tied Converters

The General Best and Worst Conditions of the Dynamic Response Due to theFundamental Frequency for the Grid-Tied Converters

To analyze the dynamic response of the grid-tied converters when theactive and reactive power is changed, the time-domain equation of the ACinput current (i_(in)(t)) which is derived during dynamic conditions isused as shown below,

$\begin{matrix}{{\Delta\;{i_{{in} - 1}(t)}} = {{c_{1}e^{{- \frac{R}{L}}t}} + \left( {{\Delta\; I_{{in} - 1 - d}{\sin\left( {\omega\; t} \right)}} + {\Delta\; I_{{in} - 1 - q}{\cos\left( {\omega\; t} \right)}}} \right)}} & (41)\end{matrix}$where ΔI_(in-1-d) and ΔI_(in-1-q) are the change of the active andreactive current that is applied to the grid-tied converter. Also, L andR are the coupling inductance and the parasitic resistance of thecoupling inductance. The term c₁ is the transient response constant forthe fundamental frequency and can be obtained by the following equation,

$\begin{matrix}{c_{1} = {- {e^{\frac{R}{L}t_{0}}\left( {{\Delta\; I_{{in} - 1 - d}{\sin\left( {\omega\; t_{0}} \right)}} + {\Delta\; I_{{in} - 1 - q}{\cos\left( {\omega\; t_{0}} \right)}}} \right)}}} & (42)\end{matrix}$where to is the time instant that ΔI_(in-1-q) and ΔI_(in-1-d) areapplied to the grid-tied converter. As analyzed in [18],[19], to have atransient-free dynamic response based on (42), when sin(ωt₀) is equal tozero, the active current (ΔI_(in-1-d)) should be changed, and whencos(ωt₀) is zero, the reactive current (ΔI_(in-1-q)) should be changed.These conditions can be summarized as,

$\begin{matrix}\left\{ \begin{matrix}\left. {\Delta\; I_{{in} - 1 - d}\mspace{14mu}{should}\mspace{14mu}{be}\mspace{14mu}{changed}}\rightarrow{k\;\pi} \right. \\\left. {\Delta\; I_{{in} - 1 - q}\mspace{14mu}{should}\mspace{14mu}{be}\mspace{14mu}{changed}}\rightarrow{{k\;\pi} + \frac{\pi}{2}} \right.\end{matrix} \right. & (43)\end{matrix}$

The above conditions are not the general condition (it is just onesolution of (42) which can be used to change the active and reactivecurrent twice in a fundamental period for the grid-tied converters.However, this reduces the speed of changing the AC current. To find asingle time instant, (42) should be equal to zero as shown in (44),

$\begin{matrix}{\frac{\Delta\; I_{{in} - 1 - q}}{\Delta\; I_{{in} - 1 - d}} = {\left. {- {\tan\left( {\omega\; t_{0}} \right)}}\rightarrow{\omega\; t_{0}} \right. = {{\tan^{- 1}\left( {- \frac{\Delta\; I_{{in} - 1 - q}}{\Delta\; I_{{in} - 1 - d}}} \right)}.}}} & (44)\end{matrix}$

From (44) it can conclude that when the active and reactive current isdecided to be changed by the controller, there is only a single timeinstant where the AC current can be changed. If the active and reactivecurrent is changed at the time instant that is derived in (44), thecurrent does not show any transient DC offset. Otherwise, a huge DCoffset is generated which can saturate the coupling inductor or causeserious issues for the controller or damage solid-state switches. Theconditions that are obtained in (43) can be derived from (44).

The solution of (44) gives the best scenario for changing the active andreactive current of the grid-tied converters. However, there is also theworst scenario for changing the active and reactive current for thegrid-tied converters. To find the worst scenario of the transientresponse of the grid-tied converter, the maximum of (42) can be foundbased on the first derivative of (42),

$\begin{matrix}{\frac{{dc}_{1}}{{dt}_{0}} = {\left. 0\Rightarrow{{{- \frac{R}{L}}{e^{\frac{R}{L}t_{0}}\left( {{\Delta\; I_{{in} - 1 - d}{\sin\left( {\omega\; t_{0}} \right)}} + {\Delta\; I_{{in} - 1 - q}{\cos\left( {\omega\; t_{0}} \right)}}} \right)}} - {e^{\frac{R}{L}t_{0}}\left( {{\Delta\; I_{{in} - 1 - d}\omega\;{\cos\left( {\omega\; t_{0}} \right)}} - {\Delta\; I_{{in} - 1 - q}\omega\;{\sin\left( {\omega\; t_{0}} \right)}}} \right)}} \right. = 0}} & (45)\end{matrix}$

By using some mathematical manipulation, the following time instant canbe found for the worst scenario of changing the active and reactivecurrent of the grid-tied converters,

$\begin{matrix}{{\omega\; t_{0}} = {\tan^{- 1}\left( \frac{\left( {{\frac{R}{L}\Delta\; I_{{in} - 1 - q}} + {\omega\;\Delta\; I_{{in} - 1 - d}}} \right)}{\left( {{{- \frac{R}{L}}\Delta\; I_{{in} - 1 - d}} + {\omega\;\Delta\; I_{{in} - 1 - q}}} \right)} \right)}} & (46)\end{matrix}$

To prove that (46) derives the worst scenario of the dynamic response ofthe grid-tied converters for the fundamental frequency, the secondderivative of (42) also should be checked as follow,

$\begin{matrix}{\frac{d^{2}c_{1}}{{dt}_{0}^{2}} = {{{- \left( \frac{R}{L} \right)^{2}}{e^{{(\frac{R}{L})}t_{0}}\left( {{\Delta\; I_{{in} - 1 - d}{\sin\left( {\omega\; t_{0}} \right)}} + {\Delta\; I_{{in} - 1 - q}{\cos\left( {\omega\; t_{0}} \right)}}} \right)}} - {\left( \frac{R}{L} \right)^{2}{e^{{(\frac{R}{L})}t_{0}}\left( {{\omega\;\Delta\; I_{{in} - 1 - d}\cos\left( {\omega\; t_{0}} \right)} - {\omega\;\Delta\; I_{{in} - 1 - q}{\sin\left( {\omega\; t_{0}} \right)}}} \right)}} - {\left( \frac{R}{L} \right)^{2}{e^{{(\frac{R}{L})}t_{0}}\left( {{\omega\;\Delta\; I_{{in} - 1 - d}\cos\left( {\omega\; t_{0}} \right)} - {\omega\;\Delta\; I_{{in} - 1 - q}{\sin\left( {\omega\; t_{0}} \right)}}} \right)}} - {{e^{{(\frac{R}{L})}t_{0}}\left( {{{- \omega^{2}}\Delta\; I_{{in} - 1 - d}{\sin\left( {\omega\; t_{0}} \right)}} - {\omega^{2}\Delta\; I_{{in} - 1 - q}{\cos\left( {\omega\; t_{0}} \right)}}} \right)}.}}} & (47)\end{matrix}$

From (47), it can conclude that the first and second lines are equal tozero based on (45). So, the second derivative of (47) when the firstderivative in (45) is equal to zero can be simplified as,

$\begin{matrix}{\frac{d^{2}c_{1}}{{dt}_{0}^{2}} = {{{- \left( \frac{R}{L} \right)}{e^{{(\frac{R}{L})}t_{0}}\left( {{\omega\;\Delta\; I_{{in} - 1 - d}{\cos\left( {\omega\; t_{0}} \right)}} - {\omega\;\Delta\; I_{{in} - 1 - q}{\sin\left( {\omega\; t_{0}} \right)}}} \right)}} - {{e^{{(\frac{R}{L})}t_{0}}\left( {{{- \omega^{2}}\Delta\; I_{{in} - 1 - d}{\sin\left( {\omega\; t_{0}} \right)}} - {\omega^{2}\Delta\; I_{{in} - 1 - q}{\cos\left( {\omega\; t_{0}} \right)}}} \right)}.}}} & (48)\end{matrix}$

Replacing ωt₀ which is obtained in (46) in (48) and using somemathematical manipulations, the following equation can be obtained forthe second derivative of c₁,

$\begin{matrix}{\frac{d^{2}c_{1}}{{dt}_{0}^{2}} = {{- \omega}\; e^{{(\frac{R}{L})}t_{0}}{\cos\left( {\omega\; t_{0}} \right)}{\left( \frac{\left( {{\frac{R}{L}\Delta\; I_{{in} - 1 - d}} - {\omega\;\Delta\; I_{{in} - 1 - q}}} \right)^{2} + \left( {{\frac{R}{L}\Delta\; I_{{in} - 1 - q}} + {\omega\;\Delta\; I_{{in} - 1 - d}}} \right)^{2}}{{\frac{R}{L}\Delta\; I_{{in} - 1 - d}} - {\omega\;\Delta\; I_{{in} - 1 - q}}} \right).}}} & (49)\end{matrix}$

As shown in (49), the second derivative of c₁ is not equal to zero whenthe first derivative in (46) is equal to zero. As shown in Table II,when the second derivative in (49) is positive and the reactive andactive current is changed based on (46), the transient DC response showsthe maximum negative DC transient offset (local minimum). Otherwise,when the second derivative in (49) is negative and the reactive andactive current is changed based on (46), the DC transient response showsthe maximum positive DC transient offset (local maximum).

Table II below shows status of the worst scenario of the dynamicresponse for the fundamental frequency of the AC current based on thesecond derivative in (49).

TABLE II Time instant of the The condition of the DC offset dynamicresponse dynamic response status -π/2≤ωt₀≤π/2$\frac{\Delta\; I_{{in} - 1 - d}}{\Delta\; I_{{in} - 1 - q}} > \frac{\omega\; L}{R}$Local maximum$\frac{\Delta\; I_{{in} - 1 - d}}{\Delta\; I_{{in} - 1 - q}} < \frac{\omega\; L}{R}$Local minimum π/2≤ωt₀≤3π/2$\frac{\Delta\; I_{{in} - 1 - d}}{\Delta\; I_{{in} - 1 - q}} > \frac{\omega\; L}{R}$Local minimum$\frac{\Delta\; I_{{in} - 1 - d}}{\Delta\; I_{{in} - 1 - q}} < \frac{\omega\; L}{R}$Local maximum

The Dynamic Response of Low Frequency Modulation Techniques

The low-frequency modulation techniques have two issues during thedynamic conditions. The first issue is the time delays of thelow-frequency modulation techniques to obtain the modulation index fromthe sinusoidal waveform by using the fast Fourier transform (FFT) asdiscussed. To solve this issue, the high-frequency modulation technique(PS-PWM) is used in for all of the cells of the CHB converter during thetransient period. However, the normal solution range of the PS-PWMtechnique is lower than the asymmetric SHCM-PWM technique. So, in someranges, the PS-PWM technique cannot control the active and reactivecurrent. To solve this issue, a hybrid PS-PWM and asymmetric SHCM-PWMtechnique is presented. To estimate the modulation index of thelow-frequency asymmetric SHCM-PWM technique when a dynamic response isrequired by the controller, the following equation which is derived fromthe phasor diagram can be used,

$\begin{matrix}{{{\Delta\; V_{{ac} - {CHB} - 1}} = {\left( {{R\;\Delta\; I_{{in} - 1 - d}^{*}} - {\omega\; L\;\Delta\; I_{{in} - 1 - q}^{*}}} \right) + {j\left( {{R\;\Delta\; I_{{in} - 1 - q}^{*}} + {\omega\; L\;\Delta\; I_{{in} - 1 - d}^{*}}} \right)}}},} & (50) \\{\mspace{79mu}{{M_{a - {new}} = {\frac{\pi{{\Delta\; V_{{ac} - {CHB} - 1}}}}{4\; V_{dc}} + M_{a - {old}}}},}} & (51) \\{\mspace{79mu}{{{\angle\left( V_{{ac} - {CHB} - 1 - {new}} \right)} = {{\angle\left( {\Delta\; V_{{ac} - {CHB} - 1}} \right)} + {\angle\left( V_{{ac} - {CHB} - 1 - {old}} \right)}}},}} & (52)\end{matrix}$where ΔI_(in-1-d)* and ΔI_(in-1-q)* are the references of the active andreactive current and M_(a-new) and M_(a-old) are the modulation indicesof the CHB converter after and before the dynamic response,respectively. Also, V_(ac-CHB-1-new) and V_(ac-CHB-1-old) are thefundamental CHB voltage after and before applying the dynamic response,respectively. Although the equations in (50-52) depend on the parametersof the passive filter, they estimate the new modulation index fastduring dynamic conditions.

The second issue of the low-frequency modulation techniques (which isalso existed for the high-frequency modulation techniques) is the DCoffset which is generated due to the low-order harmonics of the ACcurrent. For analyzing the effects of low-order current harmonics, theequations of (51) and (52) can also be rewritten for the harmonics ofΔi_(in)(t), when it is assumed that the changes of grid voltageharmonics are negligible when the dynamic response is applied to thegrid-tied converter,

$\begin{matrix}{{\Delta\;{i_{{in} - h}(t)}} = {{c_{h}e^{{- \frac{R}{L}}t}} + \left( {{\Delta\; I_{{in} - h - d}{\sin\left( {h\;\omega\; t} \right)}} + {\Delta\; I_{{in} - h - q}{\cos\left( {h\;\omega\; t} \right)}}} \right)}} & (53) \\{c_{h} = {- {e^{\frac{R}{L}t_{0}}\left( {{\Delta\; I_{{in} - h - d}{\sin\left( {h\;\omega\; t_{0}} \right)}} + {\Delta\; I_{{in} - 1 - q}{\cos\left( {h\;\omega\; t_{0}} \right)}}} \right)}}} & (54)\end{matrix}$where c_(h) is the constant for hth order of the transient component ofthe AC current and ΔI_(in-h-q) and ΔI_(in-h-d) are the changes of thehth order harmonic of the q and d components of the AC current. Asanalyzed in (54), for the hth order harmonic of the AC current, the bestscenario for changing the current harmonic is obtained when the c_(h) isequal to zero as follows,

$\begin{matrix}{{c_{h} = {\left. 0\Rightarrow{\omega\; t_{0}} \right. = {\frac{1}{h}{\tan^{- 1}\left( {- \frac{\Delta\; I_{{in} - h - q}}{\Delta\; I_{{in} - h - d}}} \right)}}}},} & (55)\end{matrix}$Also, for the worst scenario of the DC transient response due to thecurrent harmonics of the grid-tied converter, similar to (26) thefollowing equation can be obtained by deriving (34) as follows,

$\begin{matrix}{{\frac{{dc}_{h}}{{dt}_{0}} = {\left. 0\rightarrow{\omega\; t_{0}} \right. = {\frac{1}{h}{\tan^{- 1}\left( \frac{{\frac{R}{L}\Delta\; I_{{in} - h - q}} + {h\;\omega\;\Delta\; I_{{in} - h - d}}}{{{- \frac{R}{L}}\Delta\; I_{{in} - h - d}} + {h\;\omega\;\Delta\; I_{{in} - h - q}}} \right)}}}},} & (56)\end{matrix}$

The time instant that is obtained based on (56) gives the maximum andminimum DC offset of the hth order harmonic similar to (56) and (59).Similar to (59) and Table II, the following equation and Table III canbe obtained based on the second derivative of (54) for the worstscenario of the dynamic response for the AC current harmonics,

$\begin{matrix}{\frac{d^{2}c_{h}}{{dt}_{0}^{2}} = {{- h}\;\omega\; e^{{(\frac{R}{L})}t_{0}}{\cos\left( {h\;\omega\; t_{0}} \right)}{\left( \frac{\left( {{\frac{R}{L}\Delta\; I_{{in} - 1 - d}} - {h\;\omega\;\Delta\; I_{{in} - 1 - q}}} \right)^{2} + \left( {{\frac{R}{L}\Delta\; I_{{in} - 1 - q}} + {h\;\omega\;\Delta\; I_{{in} - 1 - d}}} \right)^{2}}{{\frac{R}{L}\Delta\; I_{{in} - 1 - d}} - {h\;\omega\;\Delta\; I_{{in} - 1 - q}}} \right).}}} & (57)\end{matrix}$

TABLE III Status of the worst scenario of the dynamic response for theAC current harmonics based on the second derivative in (57). Timeinstant of the The condition of the DC offset dynamic response dynamicresponse status -π/2≤hωt₀≤π/2$\frac{\Delta\; I_{{in} - h - d}}{\Delta\; I_{{in} - h - q}} > \frac{h\;\omega\; L}{R}$Local maximum$\frac{\Delta\; I_{{in} - h - d}}{\Delta\; I_{{in} - h - q}} < \frac{h\;\omega\; L}{R}$Local minimum π/2≤hωt₀≤3π/2$\frac{\Delta\; I_{{in} - h - d}}{\Delta\; I_{{in} - h - q}} > \frac{h\;\omega\; L}{R}$Local minimum$\frac{\Delta\; I_{{in} - h - d}}{\Delta\; I_{{in} - h - q}} < \frac{h\;\omega\; L}{R}$Local maximun

FIGS. 26 (a) and (b) show the time-domain waveforms of thev_(ac-CHB)(t), v_(ac-Grid)(t), and i_(in)(t) when the 3rd order currentharmonic is changed from 3+j3[A] to −3−j3[A] and the fundamentalfrequency current component is kept 7.09∠45° [A] for all of thesimulation time. The parameters of the simulation results in FIG. 26 issimilar to Table IV. However, all of the cells in FIG. 26 use the PS-PWMtechnique when the switching frequency of each switch is equal to 3.6kHz. Also, the inductance is 0.4 pu. FIGS. 26 (a) and (b) show the bestand worst scenarios for changing the 3rd order current harmonic based on(55) and (56), respectively. The maximum DC offset for the best andworst scenarios in FIGS. 26 (a) and (b) are equal to 1% and 60%,respectively. FIGS. 26 (c) and (d) show the time-domain waveforms of thev_(ac-CHB)(t), v_(ac-Grid)(t), and i_(in)(t) for the best and worstscenarios of the fundamental (from 7.09∠45[A] to 7.09∠−20 [A]) and thethird order current harmonic (from 3+j3[A] to −3−j3[A]), respectively.The maximum DC offset for the best and worst scenarios in FIGS. 26 (c)and (d) are equal to 0.5% and 111%, respectively. These simulations showthat even though the PS-PWM technique uses a high-switching frequency tochange the 3rd order current harmonic, the DC offset still is availablewhen the fundamental and the low-order current harmonic are changedbased on the worst scenarios in (46) and (56). However, in theconventional high-frequency PS-PWM technique due to eliminating almostall of the low-order current harmonics, the DC transient response cannotbe seen in the time-domain waveform. Also, changing the currentharmonics for the best scenario which is obtained based on (55) is sosimple for the PS-PWM technique than the asymmetric SHCM-PWM techniqueswhen the switching frequency is high. The low-frequency modulationtechniques have the worst DC transient dynamic response due to havinglow-order current harmonics with different magnitudes and phases fordifferent modulation indices. So, when the modulation index is changedto control the active and reactive current, a huge DC transient offsetcan be applied to the CHB converter. Also, the low-frequency modulationtechniques have the complexity of finding a solution set when the phasesand magnitudes of all low-order harmonics are controlled. To solve thisissue, one of the best solutions is modulating some of the cells with ahigh-frequency modulation technique (such as the PS-PWM) to control thelow-order current harmonics based on the best scenario which is obtainedin (35) and the rest of the cells modulate with the low-frequencymodulation technique to increase the efficiency of the converter. Thereare two main objectives in the simulation and experimental results forthe hybrid PS-PWM and asymmetric SCHM-PWM technique. First, it will beshown the hybrid technique has a high efficiency when the currentharmonics meet the requirements of IEEE 519. Also, the dynamicperformance of the grid-tied converter will be shown during the timeinstants that are found for the best and worst scenarios in (54) and(56).

FIG. 27(a) illustrates a control block diagram and FIG. 27(b) shows anindirect controller of the hybrid PS-PWM and asymmetric SHCM-PWMtechnique, according to an embodiment of the subject invention.

Referring to FIG. 27 (a), the open-loop control block is used to controlthe active and reactive current of the grid-tied converter based on(54). This control block is similar to the control block which isdiscussed before. To estimate the modulation index and the phase of theCHB converter (50)-(52) are used in the FIG. 27(b). FIG. 27(a)illustrates the control block diagram used in the simulation andexperiment to implement the disclosed transient-free dynamic responsewith the disclosed hybrid modulation technique. Referring to FIG. 27(a),the switchings of the hybrid ASHCM-PWM and PSPWM technique are appliedto control the N+P cells of the grid-tied converter. In addition, FIG.27(a) also demonstrates how DSP generates the signals based on themeasurements that are applied to the grid-tied converter.

FIG. 27(b) shows the indirect control block in the FIG. 27(a) which isused in simulation and experiment. The relationships in the blockdiagrams in FIG. 27(b) are obtained based on the equations (28), (29),(30), and (33). The block diagram in FIG. 27(b) is an open-loop controltechnique.

Simulation and Experimental Results

Simulation Verification

To verify the analysis and advantages of the hybrid PS-PWM andasymmetric SHCM-PWM technique, simulations were first conducted inMATLAB/Simulink for a grid-tied 3-cell, 7-level CHB converter. Thecircuit parameters are shown in Table IV. Three simulations wereconducted. For the hybrid modulation, one cell uses PS-PWM with 12switching transitions in a half-period as calculated. 2 cells useasymmetric SCHM-PWM with switching transitions K=10 (the switching anglesolutions were found based on (32) when K is reduced from initial 12 to10) in a half-period. The total number N_(PSPWM-ASHCM) of switchingtransitions of the CHB converter is therefore 22. For conventionalmodulation, PS-PWM is used for three cells and the total numberN_(PSPWM) of switching transitions in half period is 24(f_(s)=24×60=1440 Hz) based on (45) and (46).

TABLE IV Circuit Parameters of a Grid tied Converter in SimulationsParameter Symbol Value Line frequency f_(o) 60 Hz AC grid Voltage (RMS)V_(ac-Grid-1) 110 V Total rated power S_(total) 1.555 kVA Maximum DemandLoad (RMS) I_(L) 14.14 A Switching transitions in each half-periodN_(PSPWM-ASHCM) 22 for N = 1, P = 2, Switching transitions in eachhalf-period N_(PSPWM) 24 for N = 3, Number of H-bridge cells I  3 DC busvoltage V_(dc) 65 V Coupling inductance L 0.363 pu

In the first simulation, the average modulation index M of all cells is0.524 and because N+P=3, the total modulation index M_(a) of theconverter is 1.572. The phase of the CHB fundamental voltage is −21.23°.The active and reactive power is 1395 W+635VAR.

FIG. 28 (a) shows the time-domain waveforms of the grid-tied CHBconverter with the conventional PS-PWM and FIG. 28 (b) shows time-domainwaveforms of the hybrid PS-PWM and asymmetric SHCM-PWM, respectively.Calculations are based on waveforms of v_(ac-CHB)(t), v_(ac-Grid)(t),and i_(in)(t) and the minimum inductance of PS-PWM, when M=0.524 andθ₀=−21.23°. FIG. 28(c) shows the CHB current i_(in) harmonic spectrawith the conventional PS-PWM and FIG. 28 (d) shows the CHB currenti_(in) harmonic spectra of the hybrid PS-PWM and asymmetric SHCM-PWMrespectively. All of the harmonic spectrums in simulation andexperimental results are obtained by using the Matlab/Simulink. The redlines in FIGS. 28 (c) and (d) are the current harmonic limits of IEEE519. As shown in FIG. 28 (c), the 23rd, 25th, 29th, and 37th currentharmonics cannot meet the limits. However, as shown in FIG. 28 (d), allthe current harmonics up to the 100^(th) order meet the limits with thehybrid PS-PWM and asymmetric SHCM-PWM technique although it has asmaller number of switching transitions in each half-period than theconventional PS-PWM technique as shown in Table II. The TDD with thehybrid modulation also meets the TDD limit (TDD≤5%). It is higher thanthe TDD of the PS-PWM because the objective of (42) is to meet, not tominimize TDD. This gives the hybrid modulation technique more freedom tomitigate more number of harmonics.

FIG. 28 (e) shows the calculated minimum inductance for the conventionalPS-PWM at each harmonic to meet harmonic limits. It is shown that atleast 1.22 pu inductance is needed to meet all harmonic limits while thehybrid modulation technique uses only 0.363 pu.

In the second simulation, the average modulation index M of all cells is0.698. The total modulation index M_(a) of the CHB converter is 2.096.Also, the phase of the CHB fundamental voltage is −3.2°. The active andreactive power is 570 W+32VAR.

FIG. 29 (a) shows the time-domain waveforms of the CHB converter withthe conventional PS-PWM technique. FIG. 29 (b) shows the time-domainwaveforms of the CHB converter with the hybrid PS-PWM and asymmetricSHCM-PWM technique. Both waveforms v_(ac-CHB)(t), v_(ac-Grid)(t) arecalculated when M=0.698 and θ₀=−3.2°. As shown in FIG. 29 (b) there isan asymmetry between each half-period of the AC CHB voltage waveform.

FIG. 29 (c) shows current harmonic spectra i_(in)(t) of the CHBconverter for PS-PWM. FIG. 29 (d) shows current harmonic spectrai_(in)(t) of the CHB converter of the hybrid modulation technique. InFIG. 29 (c), the conventional PS-PWM cannot meet the limits for the23rd, 25th, 27th, and 31st harmonics. However, the hybrid modulation inFIG. 29 (d) can meet all of the current harmonic limits. The TDD of bothtechniques can meet the TDD limit of IEEE 519 in FIGS. 29 (c) and (d).It is worth to note that in FIGS. 28, 29 and 30, the actual backgroundgrid low-order voltage harmonics are considered in the v_(ac-Grid)(t).

In the last simulation results, the dynamic performance of the hybridPS-PWM and asymmetric SHCM-PWM technique is shown in FIG. 30 for thebest and worst scenarios of (54) and (56) when the input current ischanged from 7.09∠45° [A] to 5∠−45° [A]. The inductance L is 0.4 pu andthe DC link voltage of each cell is 70V.

In FIG. 30, the dynamic performance of the proposed hybrid PS-PWM andasymmetric SHCM-PWM technique was simulated for the best and worstscenarios of (54) and (56) when the input current is changed from7.62∠33.8° (A) to 6.184∠−36° (A). The inductance L is 0.484 pu and theDC link voltage of each cell is 70V. FIG. 30(a) shows the time-domainwaveforms of v_(ac-CHB)(t), v_(ac-Grid)(t), and i_(in)(t) for the worstscenario based on (56). As shown in FIG. 30(a) and Table V, the currenthas 55.4% transient DC offset. This huge transient DC offset in FIG.30(a) may damage the solid-state switches or degrade the performance ofthe controller. It should be pointed out that in the simulations,because the grid has an estimated 0.5Ω grid resistance, it damps thetransient response in the worst scenario. If the grid has higherinductance or lower resistance, the transient DC offset can besignificantly higher. FIG. 30(b) shows the time-domain waveforms of thegrid-tied CHB converter in the best scenario based on (54). Thetransient DC offset in FIG. 30(b) and Table V is around 8.2%. Condition(54) therefore guarantees small transient DC offset. Although thetransient DC offset due to the greatly mitigated harmonics still existsas condition (33) could not be simultaneously met, it is much smallerthan that in the worst scenario. FIGS. 30(c) and (d) show the currentharmonic spectra of the input currents at 7.62∠33.8° (A) to 6.184∠−36°(A), respectively. All of the current harmonics and the TDD meet therequirements of the IEEE 519.

Experimental Verification

A test 3-cell, 7-level grid-tied CHB converter with the same parametersas in Table IV is developed. In the first experiment, the averagemodulation index of all cells of the CHB converter is 0.524, and thetotal modulation index of the converter is 1.572. The phase of the CHBfundamental voltage is −21.23°. The active and reactive power is 1395W+635VAR.

FIGS. 31 (a) and (b) show the time-domain waveforms v_(ac-CHB)(t),v_(ac-Grid)(t), and i_(in)(t), when M=0.524 and θ₀=−21.23°, of thegrid-tied CHB converter with the conventional PS-PWM technique and thehybrid PS-PWM and asymmetric SHCM-PWM technique, respectively. FIGS. 31(c) and (d) show the current harmonic spectra of the CHB converter withthe conventional PS-PWM technique and the hybrid PS-PWM and asymmetricSHCM-PWM technique. The red lines in FIGS. 31 (c) and (d) are thecurrent harmonic limits of IEEE 519. As shown in FIG. 31 (c), the 19th,23rd, 25th, 29th, and 55th current harmonics cannot meet the limits ofIEEE-519 with the conventional PS-PWM. Different from the simulation inFIG. 28 (c), the TDD with the PS-PWM in FIG. 31 (c) cannot meet thelimits of IEEE 519 (TDD≤5%) due to the voltage harmonics of the actualgrid. This problem is solved by applying the asymmetric switching anglecontrol technique to the hybrid PS-PWM and asymmetric SHCM-PWMtechnique. In this technique, the magnitudes and phases of the CHBvoltage harmonics are controlled based on the grid voltage harmonics. Asa result, the TDD with the hybrid modulation can still meet the TDDlimits. As shown in Table IV and FIG. 31 (d), even with the smallernumber of switching transitions in the hybrid PS-PWM and asymmetricSHCM-PWM technique than the conventional PS-PWM technique, all thecurrent harmonics up to 100^(th) order can meet the limits.

In the second experiment, the average modulation index M of all cells is0.698 so the total modulation index M_(a) of the CHB converter is 2.096.The phase of the CHB fundamental voltage is −3.2°. The active andreactive power is 570 W+32VAR.

FIGS. 32 (a) and (b) show the time-domain waveforms v_(ac-CHB)(t),v_(ac-Grid)(t), of the CHB converter with the conventional PS-PWMtechnique and the hybrid PS-PWM and asymmetric SHCM-PWM technique, whenM=0.698 and θ₀=−3.2°. FIGS. 32 (c) and (d) show the current harmonicspectra of and i_(in)(t) of the CHB converter with the conventional andthe hybrid modulation techniques, also when M=0.698 and θ₀=−3.2°. InFIG. 32 (c), the conventional PS-PWM technique cannot meet limits forthe 17th, 23rd, 25th, 27th, and 31^(st) harmonics. However, the hybridmodulation technique in FIG. 32 (d) meets all of the current harmoniclimits. The TDDs of both techniques meet the TDD limit of IEEE 519 inFIGS. 18 (c) and (d).

In the last experiment, the dynamic performance of the proposed hybridPS-PWM and asymmetric SHCM-PWM technique is tested and compared in FIG.33 for the best and worst scenarios of (54) and (56) when the inputcurrent is changed from 7.62∠33.8° (A) to 6.184∠−36° (A). The inductanceL is 0.484 pu and the DC link voltage of each cell is around 70V. FIG.19 (a) shows the time-domain waveforms of v_(ac-CHB)(t), v_(ac-Grid)(t),and i_(in)(t) for the worst scenario of the dynamic performance based on(56). As shown in FIG. 19(a) and Table V, the current has 46% transientDC offset which is close to the DC transient in the simulation result.The estimated grid resistance is 0.5Ω and it damps the DC transient inthe worst scenario. If the grid has higher inductance or lowerresistance, the DC transient can be significantly higher. FIG. 33 (b)shows the time-domain waveforms of the grid-tied CHB converter in thebest scenario based on (54). The transient DC offset in FIG. 33 andTable V is around 6.8%. This value is also close to the simulatedtransient DC offset in FIG. 33 (b). FIGS. 33 (c) and (d) show thecurrent harmonic spectra of the input currents at 7.62∠33.8° (A) to6.184∠36° (A), respectively. All of the current harmonics and the TDDmeet the requirements of IEEE 519.

TABLE V Comparison of transient DC offsets from simulations andexperiments in FIGS. 29 and 33. Dynamic response best scenario worstscenario Simulation result 8.2% 55.4% in FIG. 29 Experimental 6.8%   46%result in Fig.

In summary, a hybrid PS-PWM and asymmetric SHCM-PWM technique isdisclosed for cascaded multilevel converters. The technique utilizesasymmetric SHCM to mitigate the harmonics generated from PS-PWM to meetharmonic limits with a smaller number of switching transitions andsmaller inductance than the conventional PS-PWM technique. Technique andguideline were developed for the designing of critical parameters, suchas coupling inductance and the number of switching transitions, for thehybrid PS-PWM and asymmetric SHCM-PWM technique. Moreover, the dynamicperformance of the grid-tied converter was discussed. As shown there arethe best and worst scenarios for changing the fundamental and harmonicsof AC current. As proven by simulation and experimental results, if theconverter could change the AC current in the best scenarios for thefundamental and harmonics, the AC current does not show any DC transientresponse. Finally, the hybrid PS-PWM and asymmetric SHCM-PWM techniquecan meet both harmonic and TDD limits of IEEE 519 with a smaller numberof switching transitions and smaller inductance than the conventionalPS-PWM technique.

It should be understood that the examples and embodiments describedherein are for illustrative purposes only and that various modificationsor changes in light thereof will be suggested to persons skilled in theart and are to be included within the spirit and purview of thisapplication.

All patents, patent applications, provisional applications, andpublications referred to or cited herein (including those in the“References” section, if present) are incorporated by reference in theirentirety, including all figures and tables, to the extent they are notinconsistent with the explicit teachings of this specification.

REFERENCES

-   [1] Khomfoi, S., and Tolbert, L. M.: ‘Multilevel Power Rectifiers’,    Power Electronics Handbook, The University of Tennessee, Department    of Electrical and Computer Engineering, Knoxville, Tenn., USA.-   [2] Dahidah, M. S. A.; Konstantinou, G.; Agelidis, V. G., “A Review    of Multilevel Selective Harmonic Elimination PWM: Formulations,    Solving Algorithms, Implementation and Applications,” in Power    Electronics, IEEE Transactions on, vol. 30, no. 8, pp. 4091-4106,    August 2015.-   [3] L. He, J. Xiong, H. Ouyang, P. Zhang and K. Zhang,    “High-Performance Indirect Current Control Scheme for Railway    Traction Four-Quadrant Converters,” in IEEE Transactions on    Industrial Electronics, vol. 61, no. 12, pp. 6645-6654, December    2014.-   [4] Watson, A. J.; Wheeler, P. W.; Clare, J. C., “A Complete    Harmonic Elimination Approach to DC Link Voltage Balancing for a    Cascaded Multilevel Rectifier,” in Industrial Electronics, IEEE    Transactions on, vol. 54, no. 6, pp. 2946-2953, December 2007.-   [5] Franquelo, L. G., Napoles, J., Guisado, R. C. P., Leon, J. I.,    and Aguirre, M. A.: ‘A Flexible Selective Harmonic Mitigation    Technique to Meet Grid Codes in Three-Level PWM Rectifiers’    Industrial Electronics, IEEE Transactions on, December 2007, vol.    54, no. 6, pp. 3022-3029.-   [6] A. Moeini, H. Zhao, and S. Wang “A Current Reference based    Selective Harmonic Current Mitigation PWM Technique for Cascaded    H-bridge Multilevel Active Rectifiers with Small Coupling    Inductance, Extended Harmonic Reduction Spectrum and the Ability to    Reduce the Harmonic Currents due to Grid Voltage Harmonics” IEEE    Transaction on Industrial Electronics, 2017.-   [7] A. Moeini, Z. Hui and S. Wang, “High efficiency, hybrid    Selective Harmonic Elimination phase-shift PWM technique for    Cascaded H-Bridge inverters to improve dynamic response and operate    in complete normal modulation indices,” 2016 IEEE Applied Power    Electronics Conference and Exposition (APEC), Long Beach, C A, 2016,    pp. 2019-2026.-   [8] H. Zhao and S. Wang, “A four-quadrant modulation technique for    Cascaded Multilevel Inverters to extend solution range for Selective    Harmonic Elimination/Compensation,” 2016 IEEE Applied Power    Electronics Conference and Exposition (APEC), Long Beach, C A, 2016,    pp. 3603-3610.-   [9] A. Moeini, H. Iman-Eini and M. Bakhshizadeh, “Selective harmonic    mitigation-pulse-width modulation technique with variable DC-link    voltages in single and three-phase cascaded H bridge inverters,” in    IET Power Electronics, vol. 7, no. 4, pp. 924-932, April 2014.-   [10] S. Wang, R. Crosier and Y. Chu, “Investigating the power    architectures and circuit topologies for megawatt superfast electric    vehicle charging stations with enhanced grid support functionality,”    Electric Vehicle Conference (IEVC), IEEE International, Greenville,    S.C., 2012, pp. 1-8.-   [11] IEEE Std 519, IEEE Recommended Practices and Requirements for    Harmonic Control in Electrical Power Systems, New York.-   [12] Reyes-Sierra, Margarita, and CA Coello Coello. “Multi-objective    particle swarm optimizers: A survey of the state-of-the-art.”    International journal of computational intelligence research 2.3    (2006): 287-308.-   [13] V. G. Agelidis, A. I. Balouktsis and M. S. A. Dahidah, “A    Five-Level Symmetrically Defined Selective Harmonic Elimination PWM    Strategy: Analysis and Experimental Validation,” in IEEE    Transactions on Power Electronics, vol. 23, no. 1, pp. 19-26,    January 2008.-   [14] Holmes, D. G. and T. A. Lipo (2003). Pulse Width Modulation for    Power Converters: Principles and Practice, John Wiley & Sons.

What is claimed is:
 1. A hybrid Cascaded H-Bridge (CHB) converter,comprising: a selective harmonic current mitigation pulse widthmodulation (SHCM-PWM) unit coupled to an input current and providing anoutput signal SW_(SHCM); a phase shift pulse width modulation (PSPWM)unit coupled to the input current and providing an output signalSW_(PS); and a CHB converter selectively coupled to the SHCM-PWM unitand the PSPWM unit, wherein the CHB converter is coupled to the SHCM-PWMunit under steady state condition and the CHB converter is coupled tothe PSPWM unit under dynamic condition; and wherein the input currentincludes an active current reference ΔI*_(in-d) and a reactive currentreference ΔI*_(in-q), and the CHB converter is selectively coupled tothe SHCM-PWM unit and the PSPWM unit based on the active currentreference ΔI*_(in-d) and the reactive current reference.
 2. The hybridCHB converter according to claim 1, wherein the CHB converter is coupledto the PSPWM unit under transient condition.
 3. The hybrid CHB converteraccording to claim 1, wherein the CHB converter is selectively coupledto the PSPWM unit in case the input current satisfies the followingFormula 1:|ΔI* _(in-d)|>0 & ωt=kπ, until ωt=(k+2)π.  Formula 1
 4. The hybrid CHBconverter according to claim 3, wherein the CHB converter is selectivelycoupled to the PSPWM unit in case the input current satisfies thefollowing Formula 2:|ΔI* _(in-d)>0 & ωt=kπ+π/2, until ωt=(k+2)π.  Formula 2
 5. The hybridCHB converter according to claim 4, wherein the CHB converter isselectively coupled to the SHCM-PWM unit in all cases where the inputcurrent does not satisfy either of Formula 1 and Formula
 2. 6. Thehybrid CHB converter according to claim 1, further comprising anindirect controller coupled to the input current and providing an outputcurrent v_(ac-CHB2) to the SHCM-PWM unit and the PSPWM unit.
 7. A hybridCascaded H-Bridge (CHB) converter, comprising: a selective harmoniccurrent mitigation pulse width modulation (SHCM-PWM) unit coupled to aninput current and providing an output signal SW_(SHCM); a phase shiftpulse width modulation (PSPWM) unit coupled to the input current andproviding an output signal SW_(PS); a modulation selector coupled to theoutput signal SW_(SHCM) of the SHCM-PWM unit and the output signalSW_(PS) of the PSPWM unit and providing an output signal SW, wherein theinput current includes an active current reference ΔI*_(in-d) and areactive current reference ΔI*_(in-q), and wherein the modulationselector selects one of the output signal SW_(SHCM) and the outputsignal SW_(PS) based on the active current reference ΔI*_(in-d) and thereactive current reference ΔI*_(in-d); and a CHB converter coupled tothe output signal SW of the modulation selector.
 8. The hybrid CHBconverter according to claim 7, wherein the modulation selector isconnected to the input current.
 9. The hybrid CHB converter according toclaim 7, further comprising an indirect controller coupled to the inputcurrent and providing an output current v_(ac-CHB2) to the SHCM-PWM unitand the PSPWM unit.
 10. The hybrid CHB converter according to claim 9,further comprising a phase lock loop (PLL) coupled to the modulationselector and an output of the CHB converter.
 11. The hybrid CHBconverter according to claim 7, wherein the modulation selector selectsthe output signal SW_(PS) in case the input current satisfies thefollowing Formulas 3 and 4:ΔI* _(in-q)>0 & ωt=kπ, until ωt=(k+2)π,  Formula 3ΔI* _(in-q)>0 & ωt=kπ+π/2, until ωt=(k+2)π.  Formula 4
 12. The hybridCHB converter according to claim 11, the modulation selector selects theoutput signal SW_(SHCM) in all cases where the input current does notsatisfy both Formula 3 and Formula
 4. 13. A hybrid Cascaded H-Bridge(CHB) converter, comprising: an asymmetric selective harmonic currentmitigation pulse width modulation (ASHCM-PWM) unit coupled to an inputcurrent and providing an output signal SW_(ASHCM); a phase shift pulsewidth modulation (PSPWM) unit coupled to the input current and providingan output signal SW_(PS); a P-cell H-Bridge coupled to the output signalSW_(ASHCM) of the ASHCM-PWM unit; and a N-cell H-Bridge coupled to theoutput signal SW_(PS) of the PSPWM unit.
 14. The hybrid CHB converteraccording to claim 13, further comprising an indirect controller coupledto the input current and providing an output current to the ASHCM-PWMunit and the PSPWM unit.
 15. The hybrid CHB converter according to claim14, wherein the input current includes an active current referenceΔI*_(in-1-d) and a reactive current reference ΔI*_(in-1-q), and theP-cell inputs the output signal SW_(ASHCM) active current referenceΔI*_(in-1-d) and the N-cell inputs the output signal SW_(PS) based onthe reactive current reference ΔI*_(in-1-q).
 16. The hybrid CHBconverter according to claim 13, wherein the P-cell H-Bridge is coupledto the ASHCM-PWM unit under steady state condition and the N-cellH-Bridge is coupled to the PSPWM unit under dynamic condition.
 17. Thehybrid CHB converter according to claim 15, wherein the active currentreference ΔI*_(in-1-d) and the reactive current reference ΔI*_(in-1-q)are limited to change once in each half period.
 18. The hybrid CHBconverter according to claim 14, wherein the P-cell H-Bridge and theN-cell H-Bridge are connected to a coupling inductance.